A Conditionally Convergent Series

Calculus Level 3

The value of the following conditionally convergent series can be expressed in the form ln ⁡ ( a ) + b \ln{(a)} + b where a a and b b are integers and the natural logarithm is fully simplified. What is the value of a + b a + b ?

∑ n = 1 ∞ ( − 1 ) n 1 n + 1 \displaystyle \sum_{n=1}^{\infty} (-1)^{n} \frac{1}{n+1}


The answer is 1.

Cole Coupland by Cole Coupland , 24, Canada

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1 solution

Mas Mus
Mar 17, 2015

∑ n = 1 ∞ ( − 1 ) n 1 n + 1 = − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + … = ( 1 − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + … ) − 1 = ln ⁡ ( 1 + 1 ) − 1 = ln ⁡ ( a ) + b \begin{aligned} \displaystyle \sum_{n=1}^{\infty} (-1)^{n} \frac{1}{n+1}& =-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\dots\\ &=(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\dots)-1\\&=\ln(1+1)-1=\ln(a)+b\end{aligned}

So, a + b = 2 − 1 = 1 a+b=2-1=1

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