Tagged Questions

0
votes
1answer
32 views

How to maximize $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?

Short Version of the Question: How do I maximize the value of $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$? Long Version of the Question: I'm currently attempting ...
1
vote
0answers
25 views

Binomial Expansion problem error

I tried solving this question but failed. a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible. b) By substituting ...
7
votes
1answer
183 views

Summation of an Infinite Series: $\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$

I am having trouble proving that $$\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$$ I know that $$\frac{2x \ \arcsin(x)}{\sqrt{1-x^2}} = \sum_{n=1}^\infty ...
8
votes
1answer
162 views

Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$

Evaluate $$\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$$ I don't understand where to start. Please help.
5
votes
2answers
67 views

Prove that $\lim_{n \to \infty} \binom{n}{k}a^n = 0$

I'm working with this problem but I have no idea how to solve it. Here $k$ is fixed and $0<a<1$. I was trying to use that $\lim_{n \to \infty} a^n =0$ and that $\binom{n}{k}\leq\frac{n^k}{k!}$ ...
5
votes
2answers
110 views

Generalization Of The Binomial Theorem

Consider the sum $$\sum_{k=0}^{n_0} {n \choose k} \cdot \alpha^k$$ where $\alpha \in \mathbb{R}$ arbritary, $n_0 < n$. So it looks like binomial theorem, $$\sum_{k=0}^n {n \choose k} \cdot ...
3
votes
5answers
221 views

Spivak's Calculus - Exercise 4.a of 2nd chapter

4 . (a) Prove that $$\sum_{k=0}^l \binom{n}{k} \binom{m}{l-k} = \binom{n+m}{l}.$$ Hint: Apply the binomial theorem to $(1+x)^n(1+x)^m$. I'm having a hard time trying to solve the problem ...
1
vote
2answers
198 views

Limit of binomial coefficient

I would like to find the limit $$ \lim_{n \to \infty} \binom{s}{n+1} = \lim_{n \to \infty} \frac{s (s-1) \cdots (s-n)}{(n+1)!} , $$ where $s \in \mathbb C$. Actually, it would be enough to show that ...
1
vote
1answer
98 views

Help with binomial theorem related proof

I'm currently working through Spivak on my own. I'm stuck on this proof, and the answer key is extremely vague on this problem. I think I'm missing a manipulation involving sums. Prove that ...
1
vote
0answers
243 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
9
votes
4answers
189 views

Bounding ${(2d-1)n-1\choose n-1}$

Claim: ${3n-1\choose n-1}\le 6.25^n$. Why? Can the proof be extended to obtain a bound on ${(2d-1)n-1\choose n-1}$, with the bound being $f(d)^n$ for some function $f$? (These numbers ...
28
votes
4answers
2k views

$n$th derivative of $e^{1/x}$

I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm ...