Question

Please (SERIES QUESTION)can someone take a look at this problems and tell me what i should know about series so i can solve these "The question is :find the interval of convergence and study it's nature" (I am not if this is the right English translation": http://sdrv.ms/1bWK318

See tomorrow i have the big test but the series was the last chapter and i haven't studied it yet

Answer 1

You can find the interval of convergence by applying the ratio test. What does this test tell you about the convergence of a series?

Answer 2

so i mainly only need the ratio test ?

Answer 3

i am not looking t solve them just what are the basic things i have to learn

Answer 4

You could also use the root test, but some curricula don't include that. But yes, for this particular problem(s?) the ratio test should suffice.

Answer 5

ok than thanks i am closing this

Answer 6

For example, consider the convergent series \(\displaystyle\sum_{n=0}^\infty \left(\frac{x}{5}\right)^n\). Because it's a geometric series, you know it converges if the common ratio of each term \(\left(\dfrac{x}{5}\right)\) is less than 1, or for \(\left|\dfrac{x}{5}\right|<1\). From this, you know that \(|x|<5\) for the series to converges, which means the interval of convergence would be \(-5<x<5\).

Let's use the ratio test to get the same conclusion:
\[\lim_{n\to\infty}\left|\frac{\left(\dfrac{x}{5}\right)^{n+1}}{\left(\dfrac{x}{5}\right)^n}\right|=\lim_{n\to\infty}\left|\frac{x}{5}\right|=\left|\frac{x}{5}\right|\]
The ratio test tells us that a series converges of this limit is less than 1, which means the series converges for \(\left|\dfrac{x}{5}\right|<1\), as expected.

Answer 7

thanks for the example too

Answer 8

You're welcome. Let me know if you're having trouble.