# Differential Calculus for the Summation of Geometric Series with Binomial Expansions

Chinnaraji Annamalai

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Chinnaraji Annamalai. Differential Calculus for the Summation of Geometric Series with Binomial Expansions. 2022. hal-03716383

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# Differential Calculus for the Summation of Geometric Series with Binomial Expansions

Chinnaraji Annamalai

School of Management, Indian Institute of Technology, Kharagpur, India

Email: anna@iitkgp.ac.in

https://orcid.org/0000-0002-0992-2584

**Abstract:** This paper presents a computational technique for finding the nth derivative of a geometric series using binomial coefficients, and some useful results. These results refer to the methodological advances which are useful for researchers who are working in computational science. Computational science is a rapidly growing multi-and inter-disciplinary area where science, engineering, computation, mathematics, and collaboration use advance computing capabilities to understand and solve the most complex real life problems.

**MSC Classification codes:** 05A10, 40A05 (65B10), 34A05

**Keywords:** computation, derivative, binomial coefficient, combinatorics

## 1. Introduction

In the earlier days, geometric series [1-5] served as a vital role in the development of differential and integral calculus and as an introduction to Taylor series and Fourier series. In calculus, the derivative of a function is the instantaneous rate of change of the function with respect to one of its independent variables. The geometric series and combinatorics [6-10] have significant applications in computational science. Computational science is a rapidly growing inter-disciplinary area where science, engineering, computation, mathematics, and collaboration use advance computing capabilities to understand and solve the most complex real life problems.

## 2. The First Derivative of Geometric Series

Differentiation is the derivative [7] of a function with respect to an independent variable. In this section, a geometric series is considered as the function of independent variable x as follows:

The function of geometric series is $f(x) = \sum_{i=0}^{r} x^i = 1 + x + x^2 + x^3 + \dots + x^r = \frac{x^{r+1} - 1}{x - 1}$.

Let $N = \{0, 1, 2, 3, 4, 5, \dots\}$ be the set of natural numbers including zero element.

The first derivative of geometric series is given below:

$$ 
\begin{aligned}
f^{1}(x) &= 1 + 2x + 3x^2 + 4x^3 + \dots + rx^{r-1} = f^{1}\left(\frac{x^{r+1} - 1}{x - 1}\right) = \frac{(rx - r - 1)x^r + 1}{(x - 1)^2} \\
\Rightarrow V_{0}^{1} + V_{1}^{1}x + V_{2}^{1}x^2 + V_{3}^{1}x^3 + \dots + V_{r-1}^{1}x^{r-1} &= \frac{(rx - r - 1)x^r + 1}{(x - 1)^2}, (x \neq 1).
\end{aligned}
$$

By substituting $x = 2$ in $f^{1}(x)$, we get the mathematical equation as follows:

$$ 1 + 2(2) + 3(2)^2 + 4(2)^3 + \dots + r2^{r-1} = \frac{(r-1)2^r + 1}{(2-1)^2} = (r-1)2^r + 1. $$
$$ \text{For } x = 3, \quad 1 + 2(3) + 3(3)^2 + 4(3)^3 + \cdots + r 3^{r-1} = \frac{(2r-1)3^r + 1}{(3-1)^2} = \frac{(2r-1)3^r + 1}{2^2}. $$

$$ \text{For } x = 4, \quad 1 + 2(4) + 3(4)^2 + 4(4)^3 + \cdots + r 4^{r-1} = \frac{(3r-1)4^r + 1}{(4-1)^2} = \frac{(3r-1)4^r + 1}{3^2}. $$

$$ \text{Similarly, for any number } k \text{ that is equal to } x, \quad \sum_{i=0}^{r-1} V_i^1 = \frac{(kr - r - 1)k^r + 1}{(k-1)^2}. $$

### 3. The $n^{th}$ Derivative of Geometric Series

$$ y = f(x) = \sum_{i=0}^{r} x^i = \frac{x^{r+1} - 1}{x - 1}. \text{ The derivatives of y are given below.} $$

$$ \frac{1}{1!} \frac{dy}{dx} = \sum_{i=0}^{r-1} V_i^1 x^i \Rightarrow \frac{1}{2!} \frac{d^2 y}{dx^2} = \sum_{i=0}^{r-2} V_i^2 x^i \Rightarrow \frac{1}{3!} \frac{d^3 y}{dx^3} = \sum_{i=0}^{r-3} V_i^3 x^i \Rightarrow \cdots \Rightarrow \frac{1}{n!} \frac{d^n y}{dx^n} = \sum_{i=0}^{r-n} V_i^n x^i. $$

The nth derivative [7] of geometric series is

$$ \frac{1}{n!} \frac{d^n y}{dx^n} = \sum_{i=0}^{r-n} V_i^n x^i = \frac{1}{n!} f^n(x) = \frac{1}{n!} f^n \left( \frac{x^{r+1} - 1}{x - 1} \right). $$

$$ \sum_{i=0}^{r-1} V_i^1 x^i = \frac{1}{1!} f^1 \left( \frac{x^{r+1} - 1}{x - 1} \right); \quad \sum_{i=0}^{r-2} V_i^2 x^i = \frac{1}{2!} f^2 \left( \frac{x^{r+1} - 1}{x - 1} \right); \quad \& \quad \sum_{i=0}^{r-3} V_i^3 x^i = \frac{1}{3!} f^3 \left( \frac{x^{r+1} - 1}{x - 1} \right) $$

are first, second, and third derivatives respectively.

### 4. Conclusion

In this article, the $n^{th}$ derivative ($n = 1, 2, 3, \dots$) of a geometric series using binomial coefficients is introduced in an innovative way. These results refer to the methodological advances which are useful for researchers who are working in science, economics, engineering, management, computation, and medicine [11].

### References

[1] Annamalai, C. (2019) Extension of ACM for Computing the Geometric Progression. *Journal of Advances in Mathematics and Computer Science*, Vol. 31(5), pp 1-3.
https://doi.org/10.9734/JAMCS/2019/v31i530125.

[2] Annamalai, C. (2017) Computational Modelling for the Formation of Geometric Series using Annamalai Computing Method. *Jñānābha*. Vol.47(2), pp 327-330.
https://zbmath.org/?q=an%3A1391.65005.

[3] Annamalai, C. (2022) Summations of Single Terms and Successive Terms of Geometric Series. SSRN4085922. http://dx.doi.org/10.2139/ssrn.4085922.

[4] Annamalai, C. (2022) Computation of Geometric Series in Different Ways. OSF Preprints. https://doi.org/10.31219/osf.io/kx7d8.
[5] Annamalai, C. (2022) Algorithmic and Numerical Techniques for Computation of Binomial and Geometric Series. *Cambridge Open Engage*. https://www.doi.org/10.33774/coe-2022-pnx53-v8.

[6] Annamalai, C. (2022) Computation for the Summation of Integers and Geometric Progression of Powers of Two. *OSF Preprints*. https://www.doi.org/10.31219/osf.io/y35ma.

[7] Annamalai, C. (2022) Differentiation and Integration of Annamalai's Binomial Expansion. SSRN4110255. http://dx.doi.org/10.2139/ssrn.4110255.

[8] Annamalai, C. (2022) Algorithmic Technique for Computation of Binomial Expansions and Geometric Series of Multiples of Powers of Two. Zenodo, https://www.doi.org/10.5281/zenodo.6787418.

[9] Annamalai, C. (2022) Computation for the Summation of Binomial Expansions and Geometric Series of Multiples of Powers of Two. *OSF Preprints*. https://www.doi.org/10.31219/osf.io/gm26e.

[10] Annamalai, C. (2022) My New Idea for Optimized Combinatorial Techniques. SSRN4078523. http://dx.doi.org/10.2139/ssrn.4078523.

[11] Annamalai, C. (2010) Application of Exponential Decay and Geometric Series in Effective Medicine. *Advances in Bioscience and Biotechnology*, Vol. 1(1), pp 51-54. https://doi.org/10.4236/abb.2010.11008.