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Solutions to math problems from MAA magazine

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Solution to math problems from MAA magazines.

AMM 11999

Evaluate:

$$\sum_{k=1}^{\infty}\frac{(-1)^{\left\lfloor\sqrt{k}+\sqrt{k+1}\right\rfloor}}{k(k+1)}$$

AMM 12003

Given an odd positive $n$, compute:

$$\sum_{k=1}^{n}\frac{\gcd(k, n)}{\cos^2\frac{k \pi}{n}}$$

AMM 12338

Prove: $$\int_0^\infty\frac{\cos\left(x\right)-1}{x\left(e^x-1\right)} dx = \frac{1}{2}\ln\left(\pi\text{ csch}\left(\pi\right)\right)$$

MM 2147

Evaluate: $$\prod_{n=2}^{\infty}\frac{n^4+4}{n^4-1}$$

MM 2154

Let $f(n)$ denote the number of ordered partitions of a positive integer $n$ such that all of the parts are odd. For example, $f(5)=5$ since 5 can be written as $5$, $3+1+1$, $1+3+1$, $1+1+3$, and $1+1+1+1+1$. Determine $f(n)$.

MM 2161

Let $x_n$ denote the number of bitstrings of length $n$ which contain neither $11$ nor $000$ as substrings. Find a recursive formula for $x_n$.

MM 2171

Evaluate the following sums in closed form.

(a) $\displaystyle{\sum_{n=0}^{\infty}\left(\cos x -1 +\frac{x^2}{2!}-\frac{x^4}{4!}+\cdots+ \left(-1\right)^{n-1}\frac{x^{2n}}{\left(2n\right)!} \right)}$

(b) $\displaystyle{\sum_{n=0}^{\infty}\left(\sin x -x +\frac{x^3}{3!}-\frac{x^5}{5!}+\cdots+ \left(-1\right)^{n-1}\frac{x^{2n+1}}{\left(2n+1\right)!} \right)}$

MM 2186

Evaluate:

$$\int_0^1\frac{\text{arctanh}\left(x \sqrt{2-x^2}\right)}{x};dx $$