<figure><img src="image_1.png" /></figure>

HAL
open science

Computing Method for the Summation of Series of
Binomial Coefficients

Chinnaraji Annamalai

► To cite this version:

Chinnaraji Annamalai. Computing Method for the Summation of Series of Binomial Coefficients. 2022. hal-03698756

HAL Id: hal-03698756

https://hal.science/hal-03698756

Preprint submitted on 19 Jun 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Computing Method for the Summation of Series of Binomial Coefficients

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 two theorems for computation of series of binomial expansions relating to the sum of geometric series whose terms are exponents of two. This combinatorial method is a methodological advance that is used in computing and computational science and engineering.

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

**Keywords:** binomial coefficient, geometric series, summation

## 1. Introduction

The computational science is a rapidly growing interdisciplinary area where combinatorics, science, engineering, mathematics, computation, and collaboration use advance computing capabilities to meet today's challenges and solve the most complex real life problems. For this purpose, a combinatorial and computing method on a geometric series is constituted for researchers who are working in the fields of science, combinatorics, computation, and management.

The factorial function of a nonnegative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$.

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

The binomial coefficient denotes that $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, where $n, r \in N$.

## 2. Binomial Theorem

In this section, a binomial theorem is introduced using the sum of geometric series whose terms are exponents of two. The binomial theorem states that sum of the summations of binomial expansions is equal to the sum of a geometric series [1-7] whose terms are the exponents of two.

**Theorem 1:** $\sum_{i=0}^{0} \binom{0}{i} + \sum_{i=0}^{1} \binom{1}{i} + \sum_{i=0}^{2} \binom{2}{i} + \sum_{i=0}^{3} \binom{3}{i} + \dots + \sum_{i=0}^{n} \binom{n}{i} = 2^{n+1} - 1$.

*Proof* for theorem 1. The binomial theorem based on both geometric series and binomial coefficients [8] is proved by step by step process as follows.

**Proof.** $\binom{0}{0} = \frac{0!}{0!} = 1 \Rightarrow \sum_{i=0}^{0} \binom{0}{i} = 2^0; \sum_{i=0}^{1} \binom{1}{i} = \binom{1}{0} + \binom{1}{1} = 1 + 1 = 2^1;$
$$ \sum_{i=0}^{2} \binom{2}{i} = \binom{2}{0} + \binom{2}{1} + \binom{2}{2} = 1 + 2 + 1 = 2^2; \sum_{i=0}^{3} \binom{3}{i} = \binom{3}{0} + \binom{3}{1} + \binom{3}{2} + \binom{3}{3} = 2^3; \dots; $$

Similarly, we can continue this process upto $n$ such that $\sum_{i=0}^{n} \binom{n}{i} = 2^n$.

When making the summations on both sides, we get

$$ \sum_{i=0}^{0} \binom{0}{i} + \sum_{i=0}^{1} \binom{1}{i} + \sum_{i=0}^{2} \binom{2}{i} + \sum_{i=0}^{3} \binom{3}{i} + \dots + \sum_{i=0}^{n} \binom{n}{i} = \sum_{i=0}^{n} 2^i, $$

where $\sum_{i=0}^{n} 2^i = \frac{2^{n+1}-1}{2-1} = 2^{n+1} - 1$ is the geometric series with exponents of two [1 - 8].

$$ \therefore \sum_{i=0}^{0} \binom{0}{i} + \sum_{i=0}^{1} \binom{1}{i} + \sum_{i=0}^{2} \binom{2}{i} + \sum_{i=0}^{3} \binom{3}{i} + \dots + \sum_{i=0}^{n} \binom{n}{i} = 2^{n+1} - 1. $$

Hence, theorem is proved.

**Theorem 2:** $\sum_{i=0}^{k} \binom{k}{i} + \sum_{i=0}^{k+1} \binom{k+1}{i} + \sum_{i=0}^{k+2} \binom{k+2}{i} + \dots + \sum_{i=0}^{k+3} \binom{k+3}{i} + \dots + \sum_{i=0}^{n} \binom{n}{i} = 2^{n+1} - 2^k,$

where $k \le n \& k, n \in N$.

*Proof* for theorem 2: The sum of a geometric series with exponents of 2 is given below:

$$ \sum_{i=k}^{n} 2^i = 2^{n+1} - 2^k. $$

$$ \text{Then, } \sum_{i=0}^{k} \binom{k}{i} + \sum_{i=0}^{k+1} \binom{k+1}{i} + \sum_{i=0}^{k+2} \binom{k+2}{i} + \dots + \sum_{i=0}^{n} \binom{n}{i} = \sum_{i=k}^{n} 2^i. $$

$$ \text{Therefore, } \sum_{i=0}^{k} \binom{k}{i} + \sum_{i=0}^{k+1} \binom{k+1}{i} + \sum_{i=0}^{k+2} \binom{k+2}{i} + \dots + \sum_{i=0}^{n} \binom{n}{i} = 2^{n+1} - 2^k. $$

### 3. Conclusion

In this article, the binomial theorem based on binomial coefficients and geometric series have been developed and these combinatorial results can be useful for researchers who are working in science, engineering, management, and medicine [9].

### References

[1] Annamalai, C. (2009) A novel computational technique for the geometric progression of powers of two. *Journal of Scientific and Mathematical Research*, Vol.3, pp 16-17.
https://doi.org/10.5281/zenodo.6642923.

[2] 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.
[3] Annamalai, C. (2015) A Novel Approach to ACM-Geometric Progression. *Journal of Basic and Applied Research International*, Vol.2(1), pp 39-40.
https://www.ikppress.org/index.php/JOBARI/article/view/2946.

[4] 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.

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

[6] Annamalai, C. (2022) Sum of Geometric Series with Negative Exponents. SSRN4088497.
http://dx.doi.org/10.2139/ssrn.4088497.

[7] Annamalai, C. (2019) Computation of Series of Series using Annamalai's Computing Model. *OCTOGON MATHEMATICAL MAGAZINE*, Vol. 27(1), pp 1-3.
http://dx.doi.org/10.2139/ssrn.4088497.

[8] Annamalai, C. (2009) A Binomial Expansion equal to Multiple of 2 with Non-Negative Exponents. SSRN4116671. http://dx.doi.org/10.2139/ssrn.4116671.

[9] 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.