Convergence and divergence of series?

Question

anonymous · Answer

$$\sum_{n=1}^{infinity} (cosn)/10^n$$

anonymous · Answer

Is this alternating or direct comparison?? I'm so confused

dani_rose · Answer

attachment

dani_rose · Answer

http://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries.aspx

anonymous · Answer

I know the basics, but I just don't know if I can find the convergence by comparing it to 1/10^n or if I have to do alternating. My book only shows alternating can be used if it's (-1) to some power or cos(npi)

ganeshie8 · Answer

Alternating series must be of form $\sum\limits(-1)^{n}a_n$, where $a_n\ge 0$.

So your series is not an alternating series.

ganeshie8 · Answer

Familiar with absolute convergence ?

anonymous · Answer

I only use it if alternating series is verified to be convergent

dani_rose · Answer

attachment

ganeshie8 · Answer

Again, your series is not an alternating series.

anonymous · Answer

In that case, what method can I use? Would direct comparison work?

ganeshie8 · Answer

For direct comparison all the terms must be nonegative. 
But cos(n) is negaitve soemtimes. 
So you cannot use direct comparison either.

ganeshie8 · Answer

Haven't you heard of "absolute convergence" before ?

anonymous · Answer

I've really only heard of converging absolutely and conditionally when used with alternating series, but never just "absolute convergence" when unrelated to it.

ganeshie8 · Answer

Check problem C http://tutorial.math.lamar.edu/Classes/CalcII/AbsoluteConvergence.aspx

anonymous · Answer

So it's $$\left| cosn \right|/10^n \le 1/10^n$$ and converges absolutely?

ganeshie8 · Answer

Fact : 

If the absolute value of the series  $\sum \limits_{n}\left| \dfrac{\cos n}{10^n}\right|$  is convergent, 
then the series without absolute value  $\sum \limits_{n} \dfrac{\cos n}{10^n}$  is also convergent.

ganeshie8 · Answer

How do you know that the series  $\sum \limits_{n}\left| \dfrac{\cos n}{10^n}\right|$  is convergent ?

anonymous · Answer

Comparison test with (1/10)^n, and (1/10) is convergent because according to geometric series test since it is larger than 0 and less than 1

ganeshie8 · Answer

Looks good! You're right, the given series converges and it is also "absolutely convergent".

anonymous · Answer

Yay! Thank you for your help!

ganeshie8 · Answer

Np :)