Question

Answer: (4)

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Answer 1

Answer:

1 / 2

Step-by-step explanation:

The idea is to express the sum as a telescoping series.

$\displaystyle\frac1{1\times3}+\frac1{3\times5}+\frac1{5\times7}+\cdots\\\\=\sum_{n=1}^\infty\frac1{(2n-1)(2n+1)}\\\\=\frac12\sum_{n=1}^\infty\left(\frac1{2n-1}-\frac1{2n+1}\right)\\\\=\frac12\left(1-\frac13+\frac13-\frac15+\frac15-\frac17+\cdots\right)\\\\=\frac12\times1\\\\=\frac12$