31 More Series

More Series What about an arithmetic series? For a series to be convergent, we must have lim n →∞ a n = 0 (Just remember that this is required but not sufficient to ensure convergence) The second famous type of series we will work with are called telescoping series . If a n = c n - c n +1 , then what does b n look like? So if c n is a convergent sequence, then X n =1 a n is a convergent series. In general, just write out the general n th partial sum and cancel as many terms as possible.
Evaluate the following telescoping series. X n =2 1 n + 1 - 1 n X n =1 2 n +1 - 2 n
X n =4 4 n 2 - 1 X n =3 ln n n + 1
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