MATH 4318

Midterm 2 | Spring '13 | Wick

Definitions and Statements of Theorems


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

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

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(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 $[a, b]$.

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True or False (1 point each)


1. If the series converges, then

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

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

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For $1 < p < ∞$ the series Converges.

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The radius of convergence of a power series is given by


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If $R > 0$ then has radius of convergence R.

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

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8. If converges, then converges too.

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

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


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Proofs


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


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

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2. (10 points) Calculate


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

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(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 that


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5. (20 points) Let be a decreasing positive-valued function. Prove that


converges if and only if

exists.

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(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.

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