Question

Someone please explain the squeeze theorem

Answer 1

the wikipedia page is actually pretty good for that definition. i've referred people to it before and they liked it

Answer 2

http://en.wikipedia.org/wiki/Squeeze_theorem

Answer 3

can you explain the integral test?

Answer 4

for convergence?

Answer 5

yes for convergence

Answer 6

sorry i ust lost electricity

Answer 7

if your series is positive and decreasing, you can treat is as a function and take the integral from 1 to infinity of the series.

if the result is not infinity, you series converges [to an undetermined value].

if the result is ± infinity, it diverges (to ± infinity).

if you have examples, i can probably explain better

Answer 8

Einfinity, k=1 (k/(k^2+1))

Answer 9

\[\sum_{1}^{\infty}\frac{ k }{ k^2 + 1 }\]
first see if its decreasing, to be able to use the integral test.

let's treat it as a function f(k). decreasing means that f(k) > f(k+1), ie, the terms keep getting smaller.

this is true f(1) > f(2) > f(3) etc. so we can do the test and take integral.
\[\int\limits_{1}^{\infty} \frac{ x }{ x^2 + 1 }dx\]
let u = x^2 +1, du = 2x dx\[\int\limits_{2}^{\infty}\frac{ 1 }{ 2 }\frac{ du }{ u } = \frac{ 1 }{ 2 } \ln|x|\left| \right|(1 \to \infty)\]\[=\frac{ 1 }{ 2 }\left( \ln(\infty) - \ln1 \right) = \infty\]
result is infinity, therefore the series diverges