Thread: Sequence
View Single Post
Old 2020-03-21, 16:12   #6
Dr Sardonicus
 
Dr Sardonicus's Avatar
 
Feb 2017
Nowhere

3,251 Posts
Default

Quote:
Originally Posted by Citrix View Post
For k=3

S(0)=0
S(1)=(1-0)/3=1/3
S(2)=(1-1/3)/3=2/9
S(3)=(1-2/9)/3=7/27
...

Hope this helps
Yes, thanks. I obviously misread the problem.

For n > 0, S(n) clearly is

\sum_{i=1}^n\frac{(-1)^{i-1}}{k^i}

which is a partial sum of a geometric series with first term 1/k and ratio -1/k.

Closed form for S(n) already given. S(n) \rightarrow 1/(k+1) for any k > 1 whether integer or not.
Dr Sardonicus is offline   Reply With Quote