- #1
Mr Davis 97
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I know the result that if ##\lim a_{2n} = L = \lim a_{2n-1}##, then ##\lim a_n = L##. I'm wondering, can this be generalized to series? i.e., if ##\displaystyle \sum_{n=1}^{\infty}a_{2n-1}## and ##\displaystyle\sum_{n=1}^{\infty}a_{2n}## converge, then ##\displaystyle \sum_{n=1}^{\infty}a_{n}## converges also? I think I can see how I would prove it using the definition of convergence, but is there a slicker way to prove it using just the algebraic properties of series?