Question

1+x+x^2+...+x^n+...
Find the interval of convergence

Answer 1

is it
-1<x<=1?

Answer 2

\[\sum_{k=0}^{\infty}x^k=\frac{1}{1-x}\iff |x|<1\]

Answer 3

strict inequality

Answer 4

so lim(1^n) is divergent

Answer 5

if x = 1 you get
\[1+1+1+1+...\] not finite

Answer 6

Using p series??????????

Answer 7

and if x = -1 you get
\[1-1+1-1+...\] which has no limit

Answer 8

If the limit oscillate we have to further investigate it dude

Answer 9

so why i have written this: \[\sum_{n=0}^{\infty}r ^{n}\]

-1<r<=1 convergent
all other r divergent

Answer 10

the inequality on the right is wrong. it must be
\[-1<r<1\] more succinctly written as
\[|r|<1\]

Answer 11

Matematika: the sequence r^n is a converging sequence when r=1, but the series isnt.

Answer 12

you cannot sum up an infinite number of "1's" that is clear yes?
\[1+1+1+...=\lim_{n\rightarrow \infty}n\]

Answer 13

to be more precise,
\[\sum_{k=0}^n1^n=n+1\] and by definition
\[\sum_{n=0}^{\infty}1^n=\lim_{n\rightarrow \infty} \sum_{n=0}^{\infty}1^n=\lim_{n\rightarrow \infty}n+1=\infty\]

Answer 14

typo in last line, should be
\[\sum_{n=0}^{\infty}1^n=\lim_{n\rightarrow \infty} \sum_{n=0}^{n}1^n=\lim_{n\rightarrow \infty}n+1=\infty\]