Sum of an Infinite Geometric Series
Calculates the sum of an infinite geometric series with first term a and common ratio r.
| \(\normalsize S_{\infty} =a+ar+ar^2+ar^3+\cdots +ar^{n-1} + \cdots = \large \frac{a}{1-r} \\ \) |
| Sum of an Infinite Geometric Series |
| \(\normalsize Geometric\ progression\\ (1)\ a_n=ar^{n-1}\\ (2)\ Sn={\large \displaystyle \sum_{\small k=1}^{\small n}}ar^{k-1}\\ \hspace{45px} =a+ar+ar^2+ar^3+\cdots +ar^{n-1}\\ \hspace{45px} =\left\{\begin{array}{1}{\large\frac{a(1-r^n)}{1-r}\hspace{40px}r\neq 1}\\ \hspace{10px} {\normalsize na\hspace{70px}r=1}\end{array}\right.\\ When -1 < r < 1, \\ \displaystyle \lim_{n \to \infty} S_{n} = \frac{a}{1-r} \\ \) |
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