MHB Evaluating Summation: Find $\sum_{i=1}^{\infty} (-1)^{i+1}f(i)$

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The discussion revolves around evaluating the infinite series $\sum_{i=1}^{\infty} (-1)^{i+1} f(i)$, where $f(n)$ is defined in terms of harmonic numbers and binomial coefficients. The initial attempts to compute $f(n)$ yield values for small integers, leading to the observation that $f(n)$ can be expressed as $\frac{1}{n}$. This suggests that the series converges to $\ln 2$, as derived through various transformations and identities involving harmonic numbers. The participants also explore the inverse binomial transform to clarify the relationship between the sums. Ultimately, the series evaluation concludes with the result $\sum_{i=1}^{\infty} (-1)^{i+1} f(i) = \ln 2$.
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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!
 
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Pranav said:
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!

Your approach is very good because is... $f(1) = H_{1} = 1$

$f(2) = 2\ H_{1} - H_{2} = 2 - 1 - \frac{1}{2} = \frac{1}{2}$

$f(3) = 3\ H_{1} - 3\ H_{2} + H_{3} = 3 - 3 - \frac{3}{2} + 1 + \frac{1}{2} + \frac{1}{3} = \frac{1}{3}$ ... and proceeding in this way You arrive at the very suggestive relation...

$\displaystyle f(n) = \sum_{k=1}^{n} (-1)^{k+1} \binom{n}{k}\ H_{k} = \frac{1}{n}\ (1)$... so that is...$\displaystyle \sum_{i=1}^{\infty} (-1)^{i + 1} f(i) = \ln 2\ (2)$

Kind regards

$\chi$ $\sigma$
 
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Switch the order of the inner sums and use $\displaystyle \sum\limits_{k \le r \le n} (-1)^r\binom{n}{r} = \frac{k (-1)^k}{n} \binom{n}{k}.$

$$\displaystyle \begin{aligned} \sum_{n \ge 1}\sum_{1 \le r \le n} \sum_{1 \le k \le r} \frac{(-1)^{n+r}}{k} \binom{n}{r} & = \sum_{n \ge 1}\sum_{1 \le k \le n} \sum_{k \le r \le n} \frac{(-1)^{n+r}}{k} \binom{n}{r} \\& = \sum_{n \ge 1}\sum_{1 \le k \le n}\frac{(-1)^{k+n}}{n} \binom{n}{k} \\& = \sum_{n \ge 1} \frac{(-1)^{n+1}}{n} \\& = \log(2).\end{aligned} $$​
 
@Prometheus

Could you give a proof of that identity?
 
chisigma said:
Your approach is very good because is... $f(1) = H_{1} = 1$

$f(2) = 2\ H_{1} - H_{2} = 2 - 1 - \frac{1}{2} = \frac{1}{2}$

$f(3) = 3\ H_{1} - 3\ H_{2} + H_{3} = 3 - 3 - \frac{3}{2} + 1 + \frac{1}{2} + \frac{1}{3} = \frac{1}{3}$ ... and proceeding in this way You arrive at the very suggestive relation...

$\displaystyle f(n) = \sum_{k=1}^{n} (-1)^{k+1} \binom{n}{k}\ H_{k} = \frac{1}{n}\ (1)$... so that is...$\displaystyle \sum_{i=1}^{\infty} (-1)^{i + 1} f(i) = \ln 2\ (2)$

Kind regards

$\chi$ $\sigma$

Awesome! :cool:

I never thought I was so close to the answer, thanks a lot chisigma! :)

Random Variable said:
There is a proof that $ \displaystyle H_{n} = \sum_{k=1}^{n}(-1)^{k-1} \binom{n}{k} \frac{1}{k} $ on Wikipedia page for the harmonic numbers.

Harmonic number - Wikipedia, the free encyclopediaThen by applying the inverse binomial transform,

$$ \frac{1}{n} = \sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} H_{k} $$Binomial Transform -- from Wolfram MathWorld
Hi Random Variable, thank you very much for your input. :)

I have never heard of inverse binomial transform before but it looks like a useful technique. I seem to have trouble figuring out how did you use it here.

We have
$$H_{n} = \sum_{k=1}^{n}(-1)^{k-1} \binom{n}{k} \frac{1}{k} $$
From the Wolfram page you quote, $b_n$ should be of the form $\displaystyle \sum_{k=0}^n (-1)^{n-k} a_k$ but I have $(-1)^{k-1}$ instead of $(-1)^{n-k}$. :confused:

Sorry if this is silly but I have never seen that representation of $H_n$ and the binomial transform. :o

Thanks!
 
@ PranavFirst let $a_{0}=b_{0} = 0 $.

Then multiply both sides of $ \displaystyle H_{n} = \sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k} $ by $(-1)^{n+1}$.

Then $ \displaystyle (-1)^{n+1} H_{n} = (-1)^{n-1} H_{n} = \sum_{k=1}^{n} (-1)^{n+k} \binom{n}{k} \frac{1}{k} = \sum_{k=1}^{n} (-1)^{n-k} \binom{n}{k} \frac{1}{k}$.

Now take the inverse binomial transform.
 
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Random Variable said:
@ PranavFirst let $a_{0}=b_{0} = 0 $.

Then multiply both sides of $ \displaystyle H_{n} = \sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k} $ by $(-1)^{n+1}$.

Then $ \displaystyle (-1)^{n+1} H_{n} = (-1)^{n-1} H_{n} = \sum_{k=1}^{n} (-1)^{n+k} \binom{n}{k} \frac{1}{k} = \sum_{k=1}^{n} (-1)^{n-k} \binom{n}{k} \frac{1}{k}$.

Now take the inverse binomial transform.

Thanks a lot Random Variable! :)
 
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