- #1

Buffu

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## Homework Statement

Find the sum of the given infinite series.

$$S = {1\over 1\times 3} + {2\over 1\times 3\times 5}+{3\over 1\times 3\times 5\times 7} \cdots $$

2. Homework Equations

2. Homework Equations

## The Attempt at a Solution

I try to reduce the denominator to closed form by converting it to a factorial.

$$\sum_{k \ge 1} {k\over {\prod^{k + 1}_{a = 1} 2a -1 }}$$

$$\sum_{k \ge 1} {k\times \prod^{k + 1}_{a = 1} 2a\over \prod^{k + 1}_{a = 1} 2a -1 \times \prod^{k + 1}_{a = 1} 2a}$$

$$\sum_{k \ge 1} {k\times 2^{k+1} \times (k+1)! \over (2(k+1))!}$$

I hit the dead end here. Although i can simply this a bit more but i still can't find a series that i can sum easily.

Please provide some hints as to how can i proceed further.

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