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