If ∣x∣<1,∣y∣<1 and x=1, then the sum to infinity of the following series (x+y)+(x2+xy+y2)+(x3+x2y+xy2+y3)+..... is
Held on 2 Sept 2020 · Verified 6 Jul 2026.
(1+x)(1+y)x+y−xy
(1+x)(1+y)x+y+xy
(1−x)(1−y)x+y−xy
(1−x)(1−y)x+y+xy
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Let $a_{1}=1$ and for $n \geqslant 1, a_{n+1}=\frac{1}{2} a_{n}+\frac{n^{2}-2 n-1}{n^{2}(n+1)^{2}}$. Then $\left|\sum_{n=1}^{\infty}\left(a_{n}-\frac{2}{n^{2}}\right)\right|$ is equal to $\_\_\_\_$.
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Let $a_{1}, \frac{a_{2}}{2}, \frac{a_{3}}{2^{2}}, \ldots, \frac{a_{10}}{2^{9}}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_{1}+a_{2}+\ldots+a_{10}=62$, then $a_{1}$ is equal to :
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