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M1P
a Define what it means to say that a sequence (an) converges to a limit l. b Prove (directly from your definition) that i
(−1)n n^2
ii
1 − 2 en en^ + 3
c Let (an) be a sequence convergent to 0 and (bn) be the sequence defined as
bn =
n
(a 1 + a 2 + ... + an) for all n ≥ 1.
Prove that bn → 0. 2 a State the Monotonic Sequence Theorem. b Prove that the sequence (an) defined as a 1 = 0.5 and
an+1 = an − a^2 n for all n ≥ 1.
is convergent and find its limit. c Let (an) be a sequence with positive terms (an > 0 for all n > 0). Suppose there exists a number r < 1 and an integer n 0 such that
an+ an
≤ r for all n ≥ n 0.
Prove that an −→ 0. [You can use without proof the result rn^ −→ 0 .]
d Using (c), or otherwise, show that
2 n n!
a Define what it means to say that a sequence (an) is Cauchy. b State the General Principle of Convergence. c Let (an) be the following sequence
an =
2 n − 1
for all n ≥ 1.
Prove that |a 2 n − an| ≥
n 4 n − 1
d Using (c), or otherwise, show that (an) is not a convergent sequence.
a Define what it means to say that a series
∑ an is convergent. b Determine whether the series
∑ an converges in each of the following cases:
i an =
3 n + 1
3 n + 4
ii an =
1 + n n^2 + 1
iii an = (−1)n
( 1 n + 1
n + 2
2 n
) . [Give reasons in each case. You may use any standard tests and results without proof, provided that you make it clear which ones you are using.]
c Let (an) and (bn) be two sequences of positive terms such that
an bn
Prove that
∑ an is convergent if and only if
∑ bn is convergent.
a Define the radius of convergence of a power series
∑ anzn^ b Find the radius of convergence of the following power series: i
∑ (n + 1)zn. ii
∑ 2 nz^2 n+1. c Let (an) be a sequence for which there exists two positive constants r and a such that |an| rn^
→ a.
Prove that the power series
∑ anzn^ has radius of convergence
r
d What is the radius of convergence of the series
∑^ ∞
n=
1 − n 2 n n + 1
zn^?
