OpenLLM-France/wikimedia huggingface dataset:
# Pinyin/Rocket vegetable

Search Pinyin rocket
Search Pinyin vegetable
Rocket vegetable is a plant that can be eaten. It is a leaf vegetable with dark green, spiky, dandelion-like leaves and a strong, spicy, peppery flavor. Its young leaves are frequently used in salads.
huǒjiàncài shì kěshíyòng zhíwù; yǒu hěnduō shēnlǜsè, jiāncháng, púgōngyīng-bān d yèzi; yǒu hěn qiáng d xīnlà hé hújiāo wèidào; nènyè jīngcháng yòngyú salad.

## Health

As in other vegetables, rocket is one of very low calorie vegetable. 100g of fresh leaves provides just 25 calories. Nonetheless, it has many vital phytochemicals, anti-oxidants, vitamins, and minerals that may immensely benefit health.
Rú qítā shūcài, huǒjiàncài shì jídī rèliàng shūcài zhīyī. 100g xīnxiān càiyè zhǐyǒu 25 calorie rèliàng. Rán'ér, tā yǒu xǔduō zhòngyào de zhíwù-huàxué-wùzhì, kàngyǎnghuàjì, vitamin hé kuàngwùzhì, kě jídàde yǒulìyú jiànkāng.
# Pinyin/Roller Skating

Roller Skating
Today, my elder brother bought me a pair of rollerskates, as well as the hat, knee pads, elbow pads, wristers, etc. The color of the whole set are pink. I quickly put on the full set, feeling cool! I learn to skate immediately, even falling are not afraid, it was fun! I feel very happy the whole day.
Lúnhuá
jīntiān, gēgē gěi wǒ mǎile yī duì lúnhuáxié, háiyǒu màozi, hùxī, hùzhǒu, hùwàn děng. Quántào dōu shì fěnhóngsè de. Wǒ hěn kuài bǎ quántào chuānshàng, gǎnjué hěn kù! Wǒ mǎshàng xuéxí huáxíng, lián shuāijiāo dōu bùpà, hěn hǎowán! Quántiān wǒ dōu yǒu yīzhǒng xīnhuānùfàng de gǎnjué.
# Pinyin/Root

Root is the part of a plant that grows under the ground, through which the plant gets water and food.
genbuh shih zhirwuh shengzhaang zaih dihxiah d buhfehn, shii zhirwuh quuder shuiifehn her shirwuh.
# Pinyin/Road

A road is a hard surface built for vehicles to travel on.
daohluh shih jiahnzhuh geei cheliahng xirngshii d jianyihng biaaomiahn.
# Pinyin/Rock

Rock is the hard solid substance that forms part of the Earth’s surface.
yarnshir shih buhfehn Dihqiur biaaomiahn d jianyihng wuhzhih.
# Pinyin/Rocket

Search Pinyin rocket
# Pinyin/Steel

Steel is a strong hard metal that is made of a mixture of iron and carbon.
gang shih tiee her tahn hercherng d jianyihng jinshuu.
# Pinyin/Stone

Search
Stone is a hard solid mineral substance that is found in the ground, often used for building.
shir'tou shih tuudih d jianyihng kuahngwuhzhih, tongcharng yohngyur jiahnzhuh.
# Pinyin/Stool

Search Pinyin stool
# Pinyin/Steroid

Special:search/steroid prefix:Pinyin/
A steroid is a chemical, either natural or man-made, which has four cycloalkane rings in its structure. Steroids are important because some of them act as a hormone in the body. Steroid hormones include female hormone, male hormone, pregnancy hormone, and vitamin D3. Prednisone is an important man-made steroid medication, used to treat inflammation. Cholesterol is a plentiful steroid in animal cells, it is important for the membranes in cells. The body uses cholesterol to make the other steroids. Steroids that are important for vertebrates are either hormones, or cholesterol.
Lèigùchún shì yīzhǒng huàxuéwù, wúlùn tiānrán de háishì rénzào de, dōu shì yóu 4 gè huánxíng de cycloalkane zǔchéng. Mǒuxiē lèigùchún zuòwéi jīsù, duì réntǐ hěn zhòngyào. Lèigùchún jīsù bāokuò cíjīsù, xióngjīsù, yùnjīsù, hé wéishēngsù D3. Prednisone shì yīzhǒng zhòngyào de rénzào lèigùchún yàowù, yòngyú zhìliáo yánzhèng. Dòngwù xìbāo de dǎngùchún shì yīzhǒng fēngfù de lèigùchún, duì xìbāomó hěn zhòngyào. Shēntǐ yòng dǎngùchún zhìzào qítā lèigùchún. Lèigùchún, wúlùn shì jīsù huò dǎngùchún, duì jǐzhuīdòngwù dōu hěn zhòngyào.
# Pinyin/Stick

Search stick
# Pinyin/Stock

A stock is a share that somebody has bought in a company or business.
guupiaoh shih moourern zaih gongsi huoh shangyeh gouhmaai d guufehn.
# Pinyin/Stomach

The stomach is part of the digestive system. It is a big sack between the oesophagus and the small intestines. The stomach is the third stage in the digestive process. It holds food after ingestion. Food in the stomach then passes through to the small intestine where most of the food's nutrition is absorbed.
Wèibù shì xiāohuà xìtǒng de yībùfèn, shì shídào hé xiǎocháng zhījiān de dà kǒudài. Wèibù shì zài xiāohuà guòchéng zhōng de dìsān jiēduàn. Shíwù shèqǔ dào wèibù, ránhòu gěi xiǎocháng xīshōu dàbùfèn de shíwù yíngyǎng.
# Pinyin/Stop

Search stop
# Pinyin/Stimulation

See also
# Pinyin/Steam

Search water
Search steam
# Sensory Neuroscience: Hearing and speech/Outer & middle ear/auditory canal

The ear canal (aka external auditory canal) is lined with epithelial cells. Particularly in the outer $1/3$, these cells secrete wax (cerumen).

## Cerumen & cilia

The cells lining the auditory canal have cilia (specialized hair) pointed out towards the pinna, which are extremely sensitive to tactile stimulation. Human ears have quite short hairs, but they serve the same purpose as those in the chinchilla pictured to the right. Try teasing a cat's hairs while they're sleeping and you're liable to lose a precious body part - the hairs are very sensitive, and for good reason: they guard against foreign bodies entering the ear.
Cerumen has some antibiotic and antifungal properties. The cells which form the lining of the canal begin their life at the centre of the eardrum (the umbo), and migrate out towards the pinna at about the same rate as the growth of your fingernails.

## Modelling the auditory canal

The auditory canal is a tube with one end closed, so it is best suited to carrying some frequencies better than others:
$f={\frac {c}{\lambda }}={\frac {343m/s}{0.024\cdot 4}}=3430Hz$ (plus harmonics, which are natural multiples of this frequency)
# Semiconductors/Transistors as Amplifiers

When the first transistor was invented, the scientists used it as an amplifier first, just to prove the mathematical concept behind it. Using transistors as amplifiers is an incredibly important use. From this model of transistors we get such things as "Transistor Radios" which rely on transistors for signal amplification. This page is going to discuss the basic mathematical model behind transistors and their use as amplifiers.

## Base, Collector, Emitter

In a typical Bipolar-Junction Transistor (BJT), there are three layers upon which it is constructed. These layers are known as the emitter, base, and collector. The emitter and collector make up most of the mass of the semiconducting layers while the base is a much thinner layer between them. Current direction depends on whether the transistor a NPN or PNP-type device. The emitter provides the charge carriers to the transistor, base controls the device current and collector collects the charge carriers.

## NPN And PNP

NPN:a n-type,a p-type,a n-type properly doped semiconductors are joined respectively in other words are sandwiched. the first n-type semiconductor is called as collector.the next p-type semiconductor is called as base.the next n-type semi- conductor is called emitter. PNP:A p-type,a n-type,a p-type properly doped semiconductors are joined respectively in other words are sandwiched. the first p-type semiconductor is called as collector.the next n-type semiconductor is called as base.the next p-type semi- conductor is called emitter.

## Amplifier Rules

The following are the amplifier rules.
input /output relation: There is no particular input-output relation because the input and output are not fixed. They vary according to the configuration of transistor. There are 3 types of transistor configurations
then by applying Kirchhoff voltage and current rule in input and output circuits relation between input and output circuits can be found depending upon configuration.
# Semiconductors/Zener Diode

A Zener diode is a type of diode that permits current to flow in the forward direction like a normal diode, but also in the reverse direction if the voltage is larger than the rated breakdown voltage or "Zener voltage". The device was named for Clarence Zener, who discovered this electrical property.
A conventional solid-state diode will not let significant current flow if reverse-biased below its reverse breakdown voltage. By exceeding the reverse bias breakdown voltage, a conventional diode is subject to high current flow due to avalanche breakdown. Unless this current is limited by external circuitry, the diode will be permanently damaged. In case of large forward bias (current flow in the direction of the arrow), the diode exhibits a voltage drop due to internal resistance. The amount of the voltage drop depends on the design of the diode.
A Zener diode exhibits almost the same properties, except the device is especially designed so as to have a greatly reduced breakdown voltage, the so-called Zener voltage. A Zener diode contains a heavily doped p-n junction allowing electrons to tunnel from the valence band of the p-type material to the conduction band of the n-type material. A reverse-biased Zener diode will exhibit a controlled breakdown and let the current flow to keep the voltage across the Zener diode at the Zener voltage. For example, a diode with a Zener breakdown voltage of 3.2 V will exhibit a voltage drop of 3.2 V if reverse biased. However, the current is not unlimited, so the Zener diode is typically used to generate a reference voltage for an amplifier stage, or as a voltage stabilizer for low-current applications.
The breakdown voltage can be controlled quite accurately in the doping process. Tolerances to within 0.05% are available though the most widely used tolerances are 5% and 10%.
Another mechanism that produces a similar effect is the avalanche effect as in the avalanche diode. The two types of diode are in fact constructed the same way and both effects are present in diodes of this type. In silicon diodes up to about 5.6 volts, the zener effect is the predominant effect and shows a marked negative temperature coefficient. Above 5.6 volts, the avalanche effect becomes predominant and exhibits a positive temperature coefficient.
In a 5.6 V diode, the two effects occur together and their temperature coefficients neatly cancel each other out, thus the 5.6 V diode is the part of choice in temperature critical applications.
Modern manufacturing techniques have produced devices with voltages lower than 5.6 V with negligible temperature coefficients, but as higher voltage devices are encountered, the temperature coefficient rises dramatically. A 75 V diode has 10 times the coefficient of a 12 V diode.
All such diodes, regardless of breakdown voltage, are usually marketed under the umbrella term of 'zener diode'.

## Further reading

* Electronics/Diodes/Zener diode
* Practical_Electronics/Diodes#Zener_diodes
# Sensory Neuroscience: Hearing and speech/Outer & middle ear

* pinna
* auditory canal
* tympanic membrane
* middle ear
* impedence mistmatch problem
* conductive hearing loss
# Semiconductors/Variable Resistance

The drain current in a MOS transistor in linear region is given by following equations.
$IDS=\mu Cox{\frac {W}{L}}(VDS(VGS-VT)-{\frac {W}{L}}VDS^{2})$
So, for a transistor, which is in linear region, as the VDS decreases the significance of the VDS squared term slowly vanishes. For significantly small VDS, the current is proportional to VDS with the proportionality factor of
$GDS=\mu Cox{\frac {W}{L}}VDS(VGS-VT)$
The resistance is given by
$RDS={\frac {1}{\mu Cox{\frac {W}{L}})(VGS-VT)}}$
Thus the resistance is controlled by the VGS. This way, a variable resistor can be implemented by the transistor.
# Sequences and Series/Infinite products

Definition (infinite product):
Let $(b_{n})_{n\in \mathbb {N} }$ be a sequence of numbers in $\mathbb {K} =\mathbb {R}$ or $\mathbb {C}$. If the limit
    $\lim _{N\to \infty }\prod _{n=1}^{N}b_{n}$
exists, it is called the infinite product of $(b_{n})_{n\in \mathbb {N} }$ and denoted by
    $\prod _{n=1}^{\infty }b_{n}$.
Proposition (necessary condition for convergence of infinite products):
In order for the infinite product
    $\prod _{n=1}^{\infty }b_{n}$
of a sequence $(b_{n})_{n\in \mathbb {N} }$ to exist and not to be zero, it is necessary that
    $\lim _{n\to \infty }b_{n}=1$.
Proof: Suppose that not $\lim _{n\to \infty }b_{n}=1$. Then there exists $\epsilon >0$ and an infinite sequence $(n_{k})_{k\in \mathbb {N} }$ such that for all $k\in \mathbb {N}$ we have $|b_{n}-1|>\epsilon$. Thus, upon denoting
    $P_{N}:=\prod _{n=1}^{N}b_{n}$,
we will have
    $|P_{n_{k}}-P_{n_{k}-1}|=|P_{n_{k}-1}||b_{n}|$.
Suppose for a contradiction that $\lim _{N\to \infty }P_{N}$ exited and was equal to $c\in \mathbb {R}$. Then when $k$ is sufficiently large, we will have
    $\left||P_{n_{k}-1}||b_{n}|-|c|\right|\geq |c|\epsilon /2$,
which is a contradiction. $\Box$
Proposition (series criterion for the convergence of infinite products):
Let $(a_{n})_{n\in \mathbb {N} }$ be a sequence of real numbers. If
    $\sum _{n=1}^{\infty }|a_{n}|<\infty$,
then
    $\prod _{n=1}^{\infty }(1+a_{n})$
converges.
Proof: $\Box$
# Sequences and Series/Power series

Proposition (identity theorem for one-dimensional power series):
Let
    $f(z):=\sum _{n=0}^{\infty }a_{n}(z-z_{0})^{n}$ and $g(z):=\sum _{n=0}^{\infty }b_{n}(z-z_{0})^{n}$
be two (complex or real) power series that converge on $B_{\epsilon }(z_{0})$ for some $\epsilon >0$. Suppose that $z_{0}$ is an accumulation point of the set $\{z\in B_{\epsilon }(z_{0})|f(z)=g(z)\}$. Then we have $a_{n}=b_{n}$ for all $n\in \mathbb {N} _{0}$.
Proof: Assume that not $a_{n}=b_{n}$ for all $n\in \mathbb {N} _{0}$. Then there exists a least $n$ (call it $n_{0}$) such that $a_{n_{0}}\neq b_{n_{0}}$. Consider the function
    $h(z):=f(z)-g(z)=\sum _{n=0}^{\infty }(a_{n}-b_{n})(z-z_{0})^{n}$,
which is defined on at least $B_{\epsilon }(z_{0})$. Since $a_{n}=b_{n}$ for $n<n_{0}$, the power series $h$ starts at $(z-z_{0})^{n_{0}}$. Therefore,
    $j(z):={\frac {h(z)}{(z-z_{0})^{n_{0}}}}=\sum _{n=n_{0}}(a_{n}-b_{n})(z-z_{0})^{n-n_{0}}$
is a well-defined function on $B_{\epsilon }(z_{0})$ which is also continuous due to the continuity of power series. Moreover,
    $j(0)=a_{n}-b_{n}\neq _{0}$,
and by continuity of $|j(z)|$, there exists a $\delta >0$ such that $|j(z)|>0$ for all $z\in B_{\delta }(z_{0})$. But by definition,
    $h(z)=(z-z_{0})^{n}j(z)$,
so that we have for $z\in B_{\delta }(z_{0})\setminus \{z_{0}\}$ that $|h(z)|=|z-z_{0}|^{n}|j(z)|>0$ and consequently $h(z)\neq 0$, and hence $f(z)\neq g(z)$. But this contradicts the assumption that $z_{0}$ was an accumulation point of $\{z\in B_{\epsilon }(z_{0})|f(z)=g(z)\}$. $\Box$
Example (falsity of the identity theorem for multi-dimensional power series):
For multi-dimensional power series, that is power series of the type
    $h(z):=\sum _{\alpha \in \mathbb {N} _{0}^{d}}a_{\alpha }(z-z_{0})^{\alpha }$ for a $z_{0}=(z_{0,1},\ldots ,z_{0,d})\in \mathbb {C} ^{d}$,
the set $\{z|h(z)=0\}$ may have $z_{0}$ as an accumulation point even when $h$ does not vanish. An easy example (which works in any dimension $d\geq 2$) is $z_{0}=0$ and
    $h(z)=z_{1}z_{2}$.
Theorem (Abel's theorem):
Let
    $\sum _{n=1}^{\infty }a_{n}z^{n}$
be a real or complex power series of convergence radius $1$, and suppose that
    $\lim _{z\to 1}\sum _{n=1}^{\infty }a_{n}z^{n}=\alpha \in \mathbb {C}$.
Then
    $\sum _{n=1}^{\infty }a_{n}=\alpha$.
{{proof|By Abelian partial summation, we have
    $\sum _{1\leq n\leq x}a_{n}z^{n}=z^{x}A(x)-\ln(z)\int _{1}^{x}A(t)z^{t}dt$
for $|z|<1$ and $x\geq 1$, where we denote as usual
    $A(x):=\sum _{1\leq n\leq x}a_{n}$.
Substituting $z=\exp(w)$, we get
    $\sum _{1\leq n\leq x}a_{n}\exp(wn)=\exp(wx)A(x)-w\int _{1}^{x}A(t)\exp(wt)dt$.
We then put $$
# Sequences and Series/Series and integration

Theorem (Abelian partial summation):
Let $(a_{n})_{n\in \mathbb {N} }$ be a sequence of complex numbers, and let $f:[1,\infty )\to \mathbb {C}$ be differentiable on $(1,\infty )$. Finally define
    $A(x):=\sum _{1\leq n\leq x}a_{n}$.
Then for $x\geq 1$ we have
    $\sum _{1\leq n\leq x}a_{n}f(n)=A(x)f(x)-\int _{1}^{x}A(t)f'(t)dt$.
Proof: If $m=\lfloor x\rfloor$, we have
    ${\begin{aligned}\int _{1}^{x}A(t)f'(t)dt&=\sum _{k=2}^{m}\int _{k-1}^{k}A(t)f'(t)dt+\int _{m}^{x}A(t)f'(t)dt\\&=\sum _{k=2}^{m}\sum _{1\leq n\leq k-1}a_{n}\int _{k-1}^{k}f'(t)dt+\sum _{1\leq n\leq m}a_{n}\int _{m}^{x}f'(t)dt\\&=\sum _{k=2}^{m}A(k-1)(f(k)-f(k-1))+A(x)f(x)-A(m)f(m).\end{aligned}}$
But
    ${\begin{aligned}\sum _{k=2}^{m}A(k-1)(f(k)-f(k-1))&=\sum _{k=2}^{m}A(k-1)f(k)-\sum _{k=2}^{m}A(k-1)f(k-1)\\&=\sum _{k=2}^{m}A(k-1)f(k)-\sum _{k=1}^{m-1}A(k)f(k)\\&=\sum _{k=2}^{m-1}f(k)(\underbrace {A(k-1)-A(k)} _{=-a_{k}})+A(m-1)f(m)-A(1)f(1).\end{aligned}}$
so that
    $\int _{1}^{x}A(t)f'(t)dt=A(x)f(x)-\sum _{k=1}^{m}a_{k}f(k)$. $\Box$
# Serbian/Appendices

* Common phrases
* Cyrylic Alphabet
# Serbian/Appendices/Common phrases
## Serbian (Slavic)

Ekavian Speech
* Serbian: српски srpski /srp-skee/ (srpskee)
* hello: здраво zdravo /zdrA-vO/ (z like in zebra) Literal translation: healthy
* good morning: добро јутро dobro jutro
* good afternoon: добар дан dobar dan
* good evening: добро вече dobro veče
* good-bye: довиђења doviđenja /dOvidʲɛnʲɑ/ Literal translation: until we see again
* please: молим molim /mO-lim/ (moleam)
* thank you: хвала hvala /hvA-la/ Literal translation: praise
* that one: то to (not as English to)
* how much?: колико? koliko? /ko-lɛ-ko/ (coleeco, co-, -co => cup)
* English: енглески engleski /ɛn-glɛ-ski/ (-skee)
* yes: да da /dA/
* no: не ne /nɛ/
* generic toast: живели! živeli! /ʒi-vɛ-li/ (zhiveli) Literal translation: live!
* sorry: извините izvinite /ezveeneete/ (formal)
* I don’t understand: не разумем ne razumem /na rAzUmem/
* Where is [the bathroom (toilet)]?: Где је тоалет? Gde je toalet? /gdɛ jɛ toAlEt/
* Do you speak English?: Да ли говорите енглески? Da li govorite engleski?
# Serbian/Dialectology

In Serbian language have four ways of pronunciation: shtokavian, kajkavian, chakavian, torlakian.

## Shtokavian dialect

In Shtokavian have three ways of pronunciation: ikavian (letter jat have change in i, Croatia), ijekavian (letter jat have change in ije or je, Bosnia and Herzegovina, Montenegro and Croatia) and ekavian (letter jat have change in e, Serbia, in few examples jat have change in i, example: nIsam (I'm not)).
Eastern Shtokavian dialects are: Šumadija-Vojvodina dialect, Zeta-south Sandžak dialect, Eastern Herzegovinian, Kosovo-Resava dialect, Smederevo-Vršac dialect, Prizren-Timok dialect, Prizren-South Morava dialect, Svrljig-Zaplanj dialect and Timok-Lužnica dialect.
Western Shtokavian dialects are: Eastern Bosnian dialect, Younger ikavian dialect or Bosnia-Dalmatia dialect and Slavonia dialect.

## Other dialects

Other dialects are: Gora dialect or Našinian dialect, Užice dialect, Crna trava dialect (this dialect have special letters, Ss (dz), f have change in v, Bulgarian semivoice ь have change in a, this dialect haven't "prelazak l u o(traverse in l to o), in this dialect have not đ, in this dialect have noniotatied jd (pođem-pojdem) ), Šatrovian (incorrect way of speaking, example: word pivo (beer) pivo-vopi, word fudbal (football) fudbal-dbalfu), Anglo-Serbian dialect, Romano-Serbian (dialect of Roma people in Serbia, Bosnia and Herzegovina and Montenegro), Bunjevac dialect and Šokac dialect.
# Sequences and Series/Multiple limits

Theorem (interchanging summation and integration):
Let $(\Omega ,{\mathcal {F}},\mu )$ be a measure space, and let $(f_{n})_{n\in \mathbb {N} }$ be a sequence of functions from $\Omega$ to $\mathbb {K} ^{d}$, where $\mathbb {K} =\mathbb {R}$ or $\mathbb {C}$. If either of the two expressions
    $\int _{\Omega }\sum _{n=1}^{\infty }\|f_{n}(\omega )\|_{\infty }\mu (d\omega )$ or $\sum _{n=1}^{\infty }\int _{\Omega }\|f_{n}(\omega )\|_{\infty }\mu (d\omega )$
converges, so does the other, and we have
    $\int _{\Omega }\sum _{n=1}^{\infty }f_{n}(\omega )\mu (d\omega )=\sum _{n=1}^{\infty }\int _{\Omega }f_{n}(\omega )\mu (d\omega )$.
Proof: Regarding the summation as integration over $\mathbb {N}$ with σ-algebra $2^{\mathbb {N} }$ and counting measure, this theorem is an immediate consequence of Fubini's theorem, given that integration and summation are defined pointwise. $\Box$
Theorem (interchanging summation and real differentiation):
Let $(f_{n})_{n\in \mathbb {N} }$ be a sequence of continuously differentiable functions from an open subset $U$ of $\mathbb {R} ^{d}$ to $\mathbb {R} ^{k}$. Suppose that both
    $\sum _{n=1}^{\infty }\|f_{n}(x)\|_{\infty }$ and $\sum _{n=1}^{\infty }\|Df_{n}(x)\|_{\infty }$
converge for all $x\in U$, and that for all $x\in U$ there exists $\delta >0$ and a sequence $(a_{n})_{n\in \mathbb {N} }$ in $\mathbb {R}$ such that
    $\sum _{n=1}^{\infty }a_{n}<\infty$ and $\forall n\in \mathbb {N} :\forall y\in {\overline {B_{\delta }(x)}}:a_{n}\geq |Df_{n}(y)|$.
Then
    $D\sum _{n=1}^{\infty }f_{n}(x)=\sum _{n=1}^{\infty }Df_{n}(x)$
for all $x\in U$.
Proof: $\Box$
# Sequences and Series/Dirichlet‒Hurwitz series

Definition (Dirichlet‒Hurwitz series):
Let $f:\mathbb {N} \to \mathbb {C}$ be a function, and let $a\in \mathbb {C} \setminus \{-1,-2,\ldots \}$. The Dirichlet‒Hurwitz series associated to $f$ and $a$ is the function of $s\in \mathbb {C}$ given by the series
    $\sum _{n=1}^{\infty }{\frac {f(n)}{(n+a)^{s}}}$.
Definition (abscissa of absolute convergence of Dirichlet‒Hurwitz series):
Let $f:\mathbb {N} \to \mathbb {C}$ be a function, and let $a\in \mathbb {C} \setminus \{-1,-2,\ldots \}$. Suppose that there exists a number $\sigma _{a}\in \mathbb {R}$ such that
    $\sum _{n=1}^{\infty }\left|{\frac {f(n)}{(n+a)^{s}}}\right|$
converges whenever $\Re s>\sigma _{a}$ and diverges whenever $\Re s<\sigma _{a}$. Then $\sigma _{a}$ is called the abscissa of absolute convergence of the Dirichlet‒Hurwitz series associated to $f$ and $a$.
Proposition (existence of abscissa of absolute convergence of Dirichlet‒Hurwitz series):
Let $f:\mathbb {N} \to \mathbb {C}$ be a function, and let $a\in \mathbb {C} \setminus \{-1,-2,\ldots \}$. Suppose that
# South Park: The Stick of Truth/Betrayal From Within

Head to the left and enter the shop. Purchase any supplies you may need, then continue left. Now you must defeat Princess Kenny. After you defeat them, they will return as a Nazi Zombie.
Continue fighting Nazi Zombie Kenny. At some point, Kenny will use one of his special attacks. The attack will depend on the buddy you are currently fighting alongside with. Mash the button to counter the attack. After the attack, your buddy will be replaced by a different one. This process will be defeated for four turns. The first buddy will be whoever you have equipped before the fight, however the next three will be completely random. Regardless of the sequence, the attacks that go with each buddy are:
* Butters: Kenny summons Death; Butters becomes Professor Chaos to blast Death away with energy
* Kyle: Kyle kicks Ike into the bottle of Nazi Zombie goo to prevent Kenny from drinking it
* Jimmy: Kenny summons Nazi Zombie rats; Jimmy lures the rats away with his flute
* Stan: Kenny summons his Nazi Zombie Unicorn; Stan decapitates it with his sword
Once you have depleted Kenny's health, defeat him once and for all by breaking the gentleman's code and farting on his balls.