Question :

Question

It is given that lim (an) n->infinity=a.

Define bn=(a1+a2+…………+an)/n

What can you say about the convergence of {bn}?

Is (a1+a2+…………+an)= lim(an) as n-->infinity or? I'm confused!

Answer 1

My gut feeling is that
consider the sequence 1,1,1,... where the limit is 1
Dividing each term by n just gives terms of the Harmonic Series, which diverges.

On the other hand, if you start with something like
a_n = e^-n cos(nπ)
then b_n =

n
∑ e^-k cos(kπ)
k=1
and the terms converge to -1/(e+1)