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#### Pranav

##### Well-known member

- Nov 4, 2013

- 428

**Problem:**

Consider a function $f(n)$ defined as:

$$f(n)=\sum_{r=1}^n (-1)^{r+1} \binom{n}{r} \left(\sum_{k=1}^r \frac{1}{k}\right)$$

Find the value of

$$\sum_{i=1}^{\infty} (-1)^{i+1}f(i)$$

**Attempt:**

I write $\sum_{k=1}^r (1/k)=H_r$.

The sum I have to evaluate is

$$f(1)-f(2)+f(3)-f(4)+\cdots$$

I tried writing down a few terms and tried to see the difference of consecutive terms....

$$f(1)=H_1$$

$$f(2)=2H_1-H_2$$

$$f(3)=3H_1-3H_2+H_3$$

$$f(4)=4H_1-6H_2+4H_3+H_4$$

....but I don't see if this helps.

Although I have posted this in the Pre-Algebra and Algebra forum, please feel free to use any Calculus approaches as I am not sure if the problem involves the use of Calculus.

Any help is appreciated. Thanks!