Download Problem Set 3 in Mathematics 120: Convergence and Divergence of Sequences and Series and more Assignments Calculus in PDF only on Docsity!
Mathematics 120 Problem Set 3 Part A ( Problems 1 - 7)
- For each of the following seven SEQUENCES, decide whether the sequence converges or diverges. If it converges, find its limit. Give reasoning to justify your answer. (a)
2 −^ n^1
(b)
3 −n
(c)
{ (^7) n (^3) + 16n − 11 4 n^3 + 26
(d)
an
, given that a 1 = 5 and an+1 = − 0. 3 an for n ≥ 1
(e)
ln
( (^) n n + 1
(f)
n + 1 −
n
(g)
{ (^100) n n!
- For each of the following five SERIES, decide whether the series converges or diverges. If the series converges, find its sum if it’s feasible to do so and otherwise find an approximation for the sum by using a partial sum.
(a)
∑^ ∞
n=
4 −^
n^1
(b)
∑^ ∞
n=
5 −n
(c)
∑^ ∞
n=
n^4 + 6n^2 + 10 3 n^4 + 27
(d)
∑^ ∞
n=
an , given that a 1 = 5 and an+1 = − 0. 3 an for n ≥ 1
(e)
∑^ ∞
n=
ln
( (^) n n + 1
- Follow the same instructions as for Problem 2. [Hint for (e): Basel Problem]
(a)
∑^ ∞
n=
n + 1
√^1
n
(b)
∑^ ∞
n=
100 n+ n!
(c) 1 2002
(d)
∑^ ∞
n=
[ 1
n + 17
− (^5
)n
]
(e)
∑^ ∞
n=
ln 2n^2
Math 120 Prob Set 3A continued
- In this problem you’ll be using the Maclaurin series for ex^ to prove that the number e is irrational (in other words, that e cannot be represented in the form pq with p and q integers). The proof will proceed by contradiction. We suppose that p and q are positive integers for which e = pq and we set
A = (q + 1)! [e − 1 − 1 −
(q + 1)!
]
(a) Explain how we know that
e − 1 − 1 −
(q + 1)!
(q + 2)!
q + 3)!
(b) Using part (a), explain how we know that A is positive. (c) Using the hypothesis relating p, q and e, explain how we know that A is an integer. (d) Explain the steps in the following calculation.
A = (q + 1)![e − 1 − 1 − 1 2!
(q + 1)!
]
q + 2
(q + 2)(q + 3)
q + 2
(q + 2)^2
q + 2
[
q + 2
(q + 2)^2
]
q + 2
1 − (^) q+2^1 = 1 q + 1
(e) Parts (b) and(c) tell us that A is a positive integer. Part (d) tells us that A <
q + 1. Explain why these results contradict each other. (This contradiction shows that our assumption must be false. In other words, e must be irrational.) (f) Now that you’ve completed the proof, summarize it as concisely as you can. Your summary can be a prose summary or a summary outline.
- For parts (a) - (d), assume that
∑^ ∞
n=
an is a convergent series with nonnegative terms.
(a) Explain how we know that there is a positive integer N such that an < 1 provided n > N.
(b) With N as in part (a), use the Direct Comparison Theorem to show that
∑^ ∞
n=N
a^2 n converges.
(c) Use part (b) to show that
∑^ ∞
n=
a^2 n also converges.
(d) Now give another proof that
∑^ ∞
n=
a^2 n converges, this time basing your argument on the Limit Com- parison Test.
