Intereting Posts

Proof of radius of convergence exponential function
$L_{p}$ distance between a function and its translation
Proving that if $f>0$ and $\int_E f =0$, then $E$ has measure $0$
How to prove this formula for Lie derivative for differential forms
Recursive Sum of Previous Term and its Inverse
How to compute $\int \sqrt{\frac{x^2-3}{x^{11}}}dx$
What's the name of the approximation $(1+x)^n \approx 1 + xn$?
Fibonacci number identity.
Improper integral about exp appeared in Titchmarsh's book on the zeta function
Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$
When does a group of dilations/scalings exist in a metric space?
Prove that free modules are projective
Solving Wave Equation with Initial Values
Non trivial Automorphism
How I can calculate $\sum_{n=0}^\infty{\frac{n}{3^n}}$?

What is the sum of this series?

$$ 1 – \frac{2}{1!} + \frac{3}{2!} – \frac{4}{3!} + \frac{5}{4!} – \frac{6}{5!} + \dots $$

- Do these series converge to the von Mangoldt function?
- A tricky infinite sum— solution found numerically, need proof
- Convergence of $\sum^\infty_{n=1} \ln(1+\frac 1 {2^n})$
- how to compute this limit
- Sum of reciprocals of numbers with certain terms omitted
- Why is a summable family at most countable?

- Prove: ${n\choose 0}-\frac{1}{3}{n\choose 1}+\frac{1}{5}{n\choose 2}-…(-1)^n\frac{1}{2n+1}{n\choose n}=\frac{n!2^n}{(2n+1)!!}$
- Why is this $0 = 1$ proof wrong?
- Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes
- Is there any known relationship between $\sum_{i=1}^{n} f(i)$ and $\sum_{i=1}^{n} \dfrac {1}{f(i)}$
- Evaluate $\sum \sqrt{n+1} - \sqrt n$
- How to prove that $\sum_{n=1}^{\infty} \frac{(\log (n))^2}{n^2}$ converges?
- Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?
- Definition of convergence of a nested radical $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$?
- Hard Definite integral involving the Zeta function
- An infinite nested radical problem

**Hint:** We have

$$e^{-x}=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\cdots.$$

Multiply both sides by $x$ and differentiate.

Alternatively, write it as:

$$1-\frac{1}{1!} +\frac{1}{2!} – \frac{1}{3!}… +\\

\left(-\frac{1}{1!}+ \frac{2}{2!} – \frac{3}{3!}…\right)$$

The first line is $e^{-1}$ and the second line, after cancelling terms, you see is $-e^{-1}$

More generally, if $$(z)_i = z(z-1)…(z-(i-1))$$ is the falling factorial, and $p(z) = a_0(z)_0 + a_1(a)_1 + … a_k(z)_k$, then:

$$\sum_{n=0}^\infty \frac{p(n)}{n!} x^n =

e^x (a_0 + a_1x + a_2x^2 + … a_k x^k)$$

In this case, $p(z) = 1 + z = (z)_0 + (z)_1$ so

$$\sum_{n=0}^\infty \frac{n+1}{n!} x^n =

e^x (1+x)$$

And, in particular, for $x=-1$, $$\sum_{n=0}^\infty \frac{(-1)^n(n+1)}{n!} = 0$$

Maybe one can do it without using power series:

$$

\begin{align}

\sum_{n=0}^{\infty}(-1)^n\frac{n+1}{n!}

&=\sum_{n=0}^{\infty}(-1)^n\frac{n}{n!}+\sum_{n=0}^{\infty}(-1)^n\frac{1}{n!}\\

&=\sum_{n=1}^{\infty}(-1)^n\frac{1}{(n-1)!}+\sum_{n=0}^{\infty}(-1)^n\frac{1}{n!}\\

&=\sum_{k=0}^{\infty}(-1)^{k+1}\frac{1}{k!}+\sum_{n=0}^{\infty}(-1)^n\frac{1}{n!}\\&=0.

\end{align}

$$

- Does an irreducible operator generate an exact $C^{*}$-algebra?
- Does the series $\sum_{n=1}^{\infty}|x|^\sqrt n$ converge pointwise? If it then what would be the sum?
- Convergence of a Harmonic Continued Fraction
- Absolutely continuous functions and the fundamental theorem of calculus
- Inequality for the p norm of a convolution
- On inequalities for norms of matrices
- How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?
- Do there exist sets $A\subseteq X$ and $B\subseteq Y$ such that $f(A)=B$ and $g(Y-B)=X-A$?
- If $f$ is an entire function with $|f(z)|\le 100\log|z|$ and $f(i)=2i$, what is $f(1)$?
- Proof that negative binomial distribution is a distribution function?
- how to find surface area of a sphere
- Integral involving Modified Bessel Function of the First Kind
- Definition of weak time derivative
- River flowing down a slope
- Prove that $H$ is a abelian subgroup of odd order