Understanding Series Convergence and Changing n=1 to n=1+k

[ptolema]

this is just a general question: do

both converge to the same limit? is there a general rule for changing n=1 to n=1+k?

 

[Sourabh N]

Convergence of a series is not affected if you add/remove finitely many terms to it. A more rigorous statement can be found in any of the many elementary books on analysis, for e.g. W. Rudin's Principles of Mathematical Analysis.

 

[ptolema]

alright, thanks! i had an intuitive idea, but i wasn't sure if there was something i had to do to a_n first. makes a lot more sense now

 

[Sourabh N]

Yes. The other way would be to replace n by m, where m = n+k (so that the range of summation changes from 0 to Inf -> k + Inf); Since m is a dummy variable, we can as well call it n.