Is there any theorem like, that the limit of the average value of an infinite series takes the same value as the original sequence?

Let lim n->infinity (an) = a be given(i.e. converges)

Then the sequence bn is defined as follows,
lim n->infinity (a1+a2+.......+an)/n

We need to comment on the convergence/divergence of bn.

So, lim n->infinity (a1+a2+.......+an)/n = lim n->infinity (an/n)= [lim n->infinity (an)]/[lim n->infinity (n)] = a/infinity ->0
==>bn converges

Or is there anything such as that the limit of the average value of an infinite series takes the same value as the original sequence?

Similar Questions
  1. For the infinite series 162 + 54 + 18 + 6 + …e) Write the series in summation notation where the lower limit is n=1. f) What
    1. answers icon 1 answer
  2. Consider the infinite geometric series∑^(∞)_(n=1) −4(1/3)^n−1 . In this image, the lower limit of the summation notation
    1. answers icon 2 answers
  3. Consider the infinite geometric series∑ ∞ n = 1 − 4 ( 1 3 ) n − 1 . In this image, the lower limit of the summation
    1. answers icon 2 answers
  4. Consider the infinite geometric series∑ ∞ n = 1 − 4 ( 1 3 ) n − 1 . In this image, the lower limit of the summation
    1. answers icon 2 answers
more similar questions