I want to calculate the series $$\sum_{n=1}^m\frac{1}{2n+1}$$ I have found out how to do this with $$\sum_{n=1}^m\frac{1}{2n-1}$$ but I cannot find this. Please help!
Calculating the sum of 1/(2n+1) from 1 to m
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sequences-and-series
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0the first one is just the second one, when $n$ is substituted by $n+1$, plus an extra term – 2017-01-03
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2How do you do the second sum? Any way other than just adding them up? – 2017-01-03
2 Answers
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If you know how to deal with the second, just note that the two are very stricly related, as: $$\sum_{n=1}^m\frac{1}{2n-1}=1+\sum_{n=1}^m\frac{1}{2n+1}-\frac{1}{2m+1}$$
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Let $S_1$ be the first sum and $S_2$ be the second sum. Then $$ S_1 = S_2 - 1+\frac{1}{2m+1} $$