# Definitions and Statements of Theorems

1. (3 points) For a sequence of real numbers state the definition for the series to converge.

2. (4 points) State the Alternating Series Theorem.

(3 points) For a sequence of functions defined on an interval [a, b], state the definition for the sequence to converge uniformly to a function $f$ on the interval $[a, b]$.

# True or False (1 point each)

1. If the series converges, then

If for all $n$ and diverges, then diverges.

If for all $n$ and converges, then converges.

For $1 < p < ∞$ the series Converges.

The radius of convergence of a power series is given by

If $R > 0$ then has radius of convergence

*R*.If has radius of convergence $R$, then $f(x)$ belongs to

8. If converges, then converges too.

9. If for all converges and p > 1 then converges.

10. If is a sequence of non-zero real numbers, then

# Proofs

1. (5 points) Suppose that for sequence of real numbers that

Prove that the series converges if and only if the series Converges.

3. (10 points) Let the radius of convergence of respectively. Suppose that there exists a such that for all we have that Prove that

(15 points) Let $a, b \in \mathbb{R}$ with $a < b$, and let be a uniformly convergent sequence of continuous real-valued functions on [

*a, b*]. Prove that5. (20 points) Let be a decreasing positive-valued function. Prove that

converges if and only if exists.

converges if and only if exists.

(20 points)

(a) Show that the series

converges for all $x \in \mathbb{R}$

(b) Recall that a function $f(x)$ is Lipschitz if there exists a positive constant $K$ so that

Show that the function is Lipschitz.

(a) Show that the series

converges for all $x \in \mathbb{R}$

(b) Recall that a function $f(x)$ is Lipschitz if there exists a positive constant $K$ so that

Show that the function is Lipschitz.

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