1. ## alternating series 1

test the convergence or divergence for sum of (cosnpie)/n^(3/4)

my work:
(cosnpie)/n^(3/4) is similar to pie/(n^3/4)
lim as n goes to infinity is (cosnpie)/n^(3/4)times(n^3/4)/pie equals infinity
so the series diverges by limit comparison test

is this right?

2. Originally Posted by twilightstr
test the convergence or divergence for sum of (cosnpie)/n^(3/4)

my work:
(cosnpie)/n^(3/4) is similar to pie/(n^3/4)
lim as n goes to infinity is (cosnpie)/n^(3/4)times(n^3/4)/pie equals infinity
so the series diverges by limit comparison test

is this right?
It actually converges:

$\displaystyle s_n=\frac{\cos\!\left(n\pi\right)}{n^{\frac34}}=\f rac{\left(-1\right)^n}{n^{\frac34}}$ starting with $\displaystyle n=1$.

Now, we need to apply the alternating series test.

First, $\displaystyle \left|s_n\right|=\left|\frac{\left(-1\right)^n}{n^{\frac34}}\right|=\frac{1}{n^{\frac3 4}}$

Since $\displaystyle s_n>s_{n+1}\implies \frac{1}{n^{\frac34}}>\frac{1}{n^{\frac34}}$ is true, all we need to do is see if $\displaystyle \lim\left|s_n\right|=0$

Thus, $\displaystyle \lim\left|s_n\right|=\lim\frac{1}{n^{\frac34}}=0$

Therefore, the series converges.