Originally Posted by

**kbartlett** (a) Explain why, when proving the convergence of a series

∑ of a(n) between n=0 and ∞ it is permissible to ignore

finitely many terms of the series.

My Answer:

Let s(n)=∑ of a(n) between k=0 and n be the nth partial

sum. The series ∑ of a(n) between k=0 and ∞

converges iff the sequence (s(n)) of partial sums converges.

Now suppose that t(n)=∑ of a(n) between k=m and n

is the partial sum formed by ignoring the first m entries

and t(n)→l∈ℝ as n→∞.

Then s(n)=a(0)+a(1)+…+a(m-1)+t(n)

and s(n)→a(0)+a(1)+…+a(m-1)+l

by continuity of the operation of adding the constant value

a(0)+a(1)+…+a(m-1). So the

original series also converges.

(b) Use (a) together with a form of the comparison test to show

that the series ∑ of 1/ between n=0 and ∞

converges.

Note: I have been told my first answer is correct, and could some also plz tell me who to post using the normal mathematical symbols, such as the sum of and a small n. Thankyou