prove-i-1-1-n-i-1-n-1-n-N-and-if-n-gt-0-n-R-is-it-right-

Question Number 193866 by liuxinnan last updated on 21/Jun/23

p r o v e

\underset{i = 1}{\sum^{+ \infty}} \frac{1}{n^{i}} = \frac{1}{n - 1} n \in N^{*}

a n d i f n > 0 \land n \in R

i s i t r i g h t ?

Answered by AST last updated on 21/Jun/23

= \frac{a}{1 - r} = \frac{\frac{1}{n}}{1 - \frac{1}{n}} = \frac{1}{n - 1} (f o r n > 1)

F o r 0 < n < 1, \frac{a}{1 - r} = \frac{1}{1 - n} \Rightarrow A s s e r t i o n i s t r u e o n l y

f o r n > 1

Commented by Tinku Tara last updated on 22/Jun/23

Relationship holds true for

∣ r ∣=∣ \frac{1}{n} ∣< 1 \Rightarrow n < - 1 or n > 1

Commented by liuxinnan last updated on 22/Jun/23

I t h i n k t h a t i s t u r e w h e n n = 1

b e c a u s e \underset{i = 1}{\sum^{+ \infty}} \frac{1}{1^{i}} = \underset{i = 1}{\sum^{+ \infty}} 1 = + \infty

\lim_{n \to 1} \frac{1}{n - 1} = + \infty

Commented by Tinku Tara last updated on 22/Jun/23

Limit does not exist for \frac{1}{n - 1} a t n = 1

\lim_{n \to 1^{-}} \frac{1}{n - 1} = - \infty

\lim_{n \to 1^{+}} \frac{1}{n - 1} = + \infty

For formula to be considered valid

equality should also hold .

Equality does not for n = 1 .

\frac{1}{n - 1} \neq \infty

Commented by liuxinnan last updated on 22/Jun/23

y e s y o u a r e r i g h t