MATH 296 PROBLEMS 14

IGOR KRIZ

Regular problems:

1. Determine if the following series converge, and if they converge absolutely:

∑_{n=1}^{∞} \frac{(−1)^{n+1}}{\sqrt{n}} (1)

∑_{n=1}^{∞} \frac{1}{n ln^{2}(n + 1)} (2)

[Use the integral criterion, and the behaviour of series with alternating signs.]

2. A function is periodic with period a if f(x + a) = f(x) for all x.

(a) If f is periodic with period a and integrable on [0, a], show that

∫_{0}^{a} f = ∫_{b}^{b+a} f for all b.

(b) Find a function f such that f is not periodic, but f′ is. [Hint: Choose a periodic g for which it can be guaranteed that ∫_{0}^{x} g is not periodic.]

(c) Suppose that f′ is periodic with period a. Prove that f is periodic if and only if f(a) = f(0).

3. Calculate

∫_{0}^{1/2} \sqrt{\frac{1 − x}{1 + x}}dx.

4. Show that the infinite series

∑_{n=0}^{∞}(\sqrt{n^{a} − 1} − \sqrt{n^{a}})

converges for a > 2 and diverges for a = 2. [Hint: (a + b)(a − b) = a^{2} − b^{2}] Challenge problems:

5. A series

∑_{n=1}^{∞} a_{n}

is called Cesaro summable with Cesaro sum ℓ if

lim_{n→∞} \frac{s_{1} + ... + s_{n}}{n} = ℓ

where s_{k} = a_{1} + ... + a_{k}.

(a) Prove that a convergent series is always Cesaro summable, and that its Cesaro sum is automatically equal to the ordinary sum.

(b) Find a series which is Cesaro summable, but not convergent in the ordinary sense.

6. Prove that every positive rational number q can be written as a finite sum of distinct numbers of the form 1/n where n is a positive integer For example,

\frac{27}{31} = \frac{1}{2} + \frac{1}{3} + \frac{1}{27} + \frac{1}{1674}.

[Use the divergence of the series ∑ 1/n and after you get close enough, keep subtracting the largest possible number 1/n.]