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Question Number 66126 by Joel122 last updated on 09/Aug/19

$$\mathrm{Is}\mathrm{there}\mathrm{any}\mathrm{formula}\mathrm{to}\mathrm{find}\mathrm{sum}\mathrm{of}$$$$1+n^2+n^4+n^6+n^8+...+n^{2k}+...$$$$\mathrm{where}n,k\in {\mathbb{Z}}^{+}$$

Answered by mr W last updated on 09/Aug/19

$$G.P.with$$$$a_0=1$$$$q=n^2$$$$a_k={\left(n^2\right)}^k$$$$S_k=\frac{n^{2(k+1)}-1}{n^2-1}forn\ne 1$$$$S_k=kforn=1$$$$$$$$\underset{k\to \mathrm{\infty }}{\lim }{S}_k=\mathrm{\infty }sincen⩾1$$

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