Is there anything known about the value of the series $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+\frac{1}{1+2+3+4+5+6}+\cdots$ ?
value of $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+\cdots$?
2 Answers
Yes, in fact, everything is known about it. Step 1: get a formula for the denominators (I assume the 6th one is supposed to have a 6 in it, not stop at 5 like the 5th one). Step 2: apply partial fractions to get a telescoping series. Enjoy.
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1@Gerry I think the last word was meant to be "profit". – 2012-04-18
The denominators $a_n$ are $ a_n = \sum_{k=1}^{n}k = \frac{n(n+1)}{2} $ so $ \sum_{n=1}^{\infty} \frac1{a_n} = \sum_{n=1}^{\infty} \frac{2}{n(n+1)} = 2\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right) = 2 $ If you don't have the resources to check this on a computer (out to infinity), you will have your computer or calculator carry out the calculation for $n
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0whats your email address please, im not smart enough to figure it out from your profile – 2012-11-21