Question

Question about series (taking the summation): So I really don't know how to start these problems. Do I always start by seeing if it converge or diverge. and if it does converge.. i should try making them like the theorem/series types? like geometric or harmonic? or is there any easier way to do this?

Answer 1

it actually depends on the series :)

-If it's harmonic, then directly say it diverges since harmonic series always diverge, you can take values to check.

- If t's geometric series, then you have the following rule :

Answer 2

\[\frac{ar^k}{1-r}\] ; this equation is used if and only if the series is diverging to find the summation, and you only use it when |r|< 1 ( converge)

but if the |r| > 1 , then it's diverging.

Answer 3

i'm sorry, if and only if the series is converging*

Answer 4

example :\[\sum_{k=0}^{\infty} 2(\frac{-1}{2})^k\]
discuss the divergence and convergence of the following series. If it's convergent, then what's the sum?

Solution:

First find r:

R( is always the one that has the power k) = -1/2

now let's check if |r| < 1:
|-1/2| = 1/2 < 1 , so the series is converging.

Now since it's converging we have to find the sum using the following:


\[\frac{ar^k}{1-r}\]

where the numerator is the first term. Substitute K with 0 and you'll getthe first term which is = 2

so :
\[\frac{ar^k}{1-r} = \frac{2}{1+\frac{1}{2}} = \frac{4}{3}\]

so the sum of the series is 4/3


I hope this example clarified some points ^_^