Sum of $\frac{1}{1 \cdot 2} + \frac{1}{3\cdot 4} + \cdots$

calculussequences-and-series

I am practicing for the GRE, and came across the following question: Find the sum
$$
\frac{1}{1 \cdot 2} + \frac{1}{3\cdot 4} + \frac{1}{5\cdot 6}\cdots.
$$
The answer is given to be $\log(2)$, with the hint: "Apply partial fractions to each term and then recognize the series for $\log(1 + x)$ or estimate." I have no idea what partial fractions means in this context, and have tried without success to manipulate this sum by pulling out terms to get it into a familiar form. Any help is appreciated.

Best Answer

We have that

$$\sum_{k=1}^n\frac1{(2k-1)2k}=\sum_{k=1}^n \left(\frac1{2k-1}-\frac1{2k} \right)=1-\frac12+\frac13-\frac14+\ldots=\sum_{k=1}^n (-1)^{k+1}\frac1k$$

then refer to the Alternating harmonic series and the related Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $.

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