1
vote
3answers
137 views

Multiplication of infinite series

Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have ...
7
votes
1answer
204 views

Infinite series

$$\log2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots$$ $$\frac{\log2}{2}=\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}-\frac{1}{12}+\cdots$$ Adding these two ...
3
votes
2answers
91 views

What am I doing wrong?

I am trying to prove the integral test for series, but got a strange result. Assume that $f$ is decreasing and positive. Because the series can be imagined as the area-sum of $1$-wide rectangles of ...
5
votes
3answers
200 views

Error in finding sum of $1\cdot 2+3\cdot 4+ \cdots \text{to}\space n\space \text{terms}$

To find sum of the series $1\cdot 2+3\cdot 4+ \cdots \text{to}\space n\space \text{terms}$ My approach, Let S=$1^2+2^2+3^2 + \cdots +n^2$ If $n$ is even S=$(1-2)^2+(3-4)^2+ \cdots +[(n-1)-n]^2+2(1 ...
7
votes
1answer
92 views

Prove $\frac {1}{\cos 0^\circ \cdot \cos 1^\circ} + \ldots +\frac {1}{\cos 88^\circ \cdot \cos 89^\circ}= \frac{\cos 1^\circ}{\sin 1^\circ}$

Prove the following identity: $$\frac {1}{\cos 0^{\circ} \cdot \cos 1^{\circ}} + \ldots +\frac {1}{\cos 88^{\circ} \cdot \cos 89^{\circ}} = \frac{\cos 1^{\circ}}{\sin 1^{\circ}}$$ After hours of ...
11
votes
4answers
582 views

Does $\lim \frac {a_n}{b_n}=1$ imply $\lim \frac {f(a_n)}{f(b_n)}=1$?

I wanted to prove the seemingly simple statement: If $\lim \frac {a_n}{b_n}=1$ and $f$ continuous with $f(b_n)\neq0$ then $\lim \frac {f(a_n)}{f(b_n)}=1.$ I started promptly with \begin{align} ...
11
votes
2answers
442 views

What is wrong with this fake proof that $\lim\limits_{n\rightarrow \infty}\sqrt[n]{n!} = 1$?

$$\lim_{n\rightarrow \infty}\sqrt[n]{n!}=\lim_{n\rightarrow \infty}\sqrt[n]{1}*\sqrt[n]{2}\cdots\cdot\sqrt[n]{n}=1\cdot1\cdot\ldots\cdot1=1$$ I already know that this is incorrect but I am wondering ...
4
votes
1answer
282 views

proof that $0 = \infty$

I have constructed a proof of $0 = \infty$ that I know is incorrect, although I'm not quite sure why. it goes like this: $$0 = 0 = (1-1) + (1-1) + (1-1) + ...$$ but it is also true that $0. ...
6
votes
2answers
4k views

Problem when integrating $e^x / x$.

I made up some integrals to do for fun, and I had a real problem with this one. I've since found out that there's no solution in terms of elementary functions, but when I attempt to integrate it, I ...
0
votes
4answers
179 views

If $\displaystyle \sum_{n=0}^\infty c_n4^n$ is convergent, is $\displaystyle \sum_{n=0}^\infty c_n(-4)^n$ convergent as well?

Please identify the flaw in my reasoning: $\displaystyle \sum_{n=0}^\infty c_n4^n$ is convergent, so by the ratio test: $\displaystyle \lim_{n \to \infty}\left\vert\frac{a_{n+1}}{a_n}\right\vert = ...
5
votes
2answers
189 views

Fake proof of the limit of a series

Now, I know this to be correct: $$\begin{align*} \lim_{n \rightarrow\infty} \left(\frac 1{n^2}+\frac 2{n^2}+\ldots+\frac n{n^2}\right)&=\lim_{n \rightarrow\infty} \left[\frac 1{n^2} \left(\frac ...
-1
votes
1answer
182 views

How to find the area. Linked with another question. [duplicate]

Possible Duplicate: Is value of $\pi = 4$? In this question we discussed why the fake proof is wrong. But, what about the area? The process converges to the same area of the circle ...
44
votes
7answers
11k views

Infinity = -1 paradox

I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1: Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 ...