What is the sum from n=1 to infinity of (cos(n*pi))/(n^{3/8)} ?

The sum from n=1 to infinity of (cos(n*pi))/(n^{3/8)} is converges

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sum from n=1 to infinity of (cos(n*pi))/(n^{3/8)}

n = 1 \sum \infty \frac{cos ( n \cdot π )}{n ^{\frac{3}{8}}}

converges

Solution steps

n = 1 \sum \infty \frac{cos ( nπ )}{n ^{\frac{3}{8}}}

Use the basic trigonometric identity:

cos (nπ) = (- 1)^{n}

= n = 1 \sum \infty \frac{( - 1 ) ^{n}}{n ^{\frac{3}{8}}}

Apply Alternating Series Test:

converges

= converges

x \to 0 lim (\frac{sin ( π )}{x})

\frac{d}{d x} (cos (x^{2} y))

y^{^{'}} = \frac{x y + y ^{2}}{x ^{2}}

y^{^{''}} + 4 y^{^{'}} + 4 y = (7 + x) e^{- 2 x}

\frac{d}{d x} (\frac{1}{x} + \frac{2}{x ^{2}})

What is the sum from n=1 to infinity of (cos(n*pi))/(n^{3/8)} ?
The sum from n=1 to infinity of (cos(n*pi))/(n^{3/8)} is converges