All Questions
96
questions
6
votes
3
answers
750
views
Preventing "proof by homework"?
I am doing problem 3d in the Prologue of Spivak:
Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$
I feel like my proof could pass off as ...
- 3,076
2
votes
2
answers
66
views
Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$
I need to upperbound the sum
$$\sum_{n=n_1}^\infty x^n (n+1)^2$$
where $0<x<1$ is a parameter.
I know it can be done starting from
$$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n (n+...
- 1,108
1
vote
1
answer
102
views
Why is this summation formula wrong?
This is the alternate form of the summation formula:
$$
\sum^{n}_{k=0} a(c)^k = \frac{ac^{n+1} - a}{c - 1}
$$
so why is this wrong?
$$
\sum^{n}_{k=0} (-\frac{1}{2})^k = \frac{(-\frac{1}{2})^{n+1} - ...
- 1,085
1
vote
1
answer
94
views
Summation Solution differing from Integral Solution
I know this is elementary stuff but I'm hoping to clear get it cleared up.
I have a money making machine. I turn on the machine and for the first day I make $5.70.
For every subsequent day, I make ...
- 113
3
votes
2
answers
265
views
A limit about euler's constant
Show that :
$$\lim_{m\to \infty}\left[ -\frac{1}{2m}+\ln \left( \frac{\text{e}}{m} \right)+\sum\limits_{n=2}^m \left( \frac{1}{n}-\frac{\zeta \left( 1-n \right)}{m^n} \right) \right]=\gamma $$
How to ...
- 3,955
6
votes
3
answers
1k
views
Explain why calculating this series could cause paradox?
$$\ln2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - 2(\frac{1}{2} + \frac{1}{4} + \cdots)$$
$$= (1 + \frac{1}{2} + \frac{...
- 8,038