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# Compute the sum of the series ∑ (-1)n x2n

Find all such that the series

converges and compute the sum.

We have

by the expansion for the geometric series. This is convergent for .

# Prove some formulas for given infinite sums

Using the formula

prove that:

1. .
2. .

1. Proof. Taking in the given formula we have

2. Proof. Starting with the given equation we have for ,

Take , then we have

# Prove an integral formula for ∑ (sin (nx)) / n2

Prove that the series

converges for all and let denote the value of this sum for each . Prove that is continuous for and prove that

Proof. First, the series converges for all real by the comparison test since

for all . Therefore, the convergence of implies the convergence of . Furthermore, this convergence is uniform by the Weierstrass -test with given by , and again converges. Thus, by Theorem 11.2 (page 425 of Apostol),

is continuous on the interval . Therefore, we may apply Theorem 11.4 (page 426 of Apostol):

since if and equals 2 if

# Prove that the improper integral ∫ f(x) dx and ∑ f(n) both converge or both diverge

1. Assume that is a monotonically decreasing function for all and that

Prove that the improper integral and the series

both converge or both diverge.

2. Give a counterexample to the theorem in part (a) in the case that is not monotonic, i.e., find a non-monotonic function such that converges but diverges.

Incomplete.

# Show that the limit of ∑ 1/k = log (p/q) where the sum is from k = qn to pn

1. For given integers and with , prove

2. Consider the series

This is a rearrangement of the alternating harmonic series () in which there are three positive terms followed by two negative terms. Prove that the series converges and that the sum is equal to .

Incomplete.

# Prove that the sum of reciprocals of integers with no zeros in their decimal representation converges

Consider the positive integer with no zeros in their decimal representation:

Prove that the series

converges. Further, prove that the sum is less than 90.

Incomplete.

# Find integers a such that ∑ (n!)3 / (an)! converges

Find all positive integers such that the series

converges.

We use the ratio test,

Since the largest power of in the denominator is , we have that this limit diverges for . If then the limit is , and the limit is 0 if . Thus, the series converges for all .

# Find real c such that ∑ (n!)c / (3n)! converges

Find all such that the following series is convergent:

Consider the ratio test,

This limit is 0 if and is if . If then the limit diverges. Hence, by the ratio test we have that the series converges for and diverges for .

# Prove relationships between a given series

Let be a sequence of positive terms.

1. Prove or give a counterexample: if diverges then also diverges.
2. Prove or give a counterexample: if converges then also converges.

1. Counterexample. Let . Then

diverges. Furthermore, and

converges.

2. Proof. We know from a previous exercise (Section 10.20, Exercise #51) we know

for . So, for this case define . Then converges by hypothesis. Furthermore, (since so ),

Since this is the case , we have proved the statement

# Determine the convergence of ∑ ( (na + 1)1/2 – (na)1/2

Prove that if then the series

converges, and that it diverges if .

Proof. We have

Then, we use the limit comparison test with the series .

Thus, the given series converges or diverges as does. But, we know converges for and diverges for as desired