# Is there a systematic way to find the gerneral formula of partial sum of a series?

Recently, I was working on Principles of real analysis by Rudin. When I got into the topic of convergence of series (Ch3), the book said if partial sum of the series converge, then the series converge.

And, an idea popped into my head that if there is a systematic way to find out the formula of the partial sum, like geometric series does, so I can test the convergence of the series simply by showing that the limit of the formula and does not need to use so many tests, like comparison test, ratio test, root test, etc.

• This is no easy task, bro. This is the definition of a series, isn't it? – Shoutre Nov 20 '15 at 3:47
• Sometimes there is, in general there isn't. When there is, then yes of course you can try to reason about the expression that you manipulate the sum into. – BrianO Nov 20 '15 at 3:48
• Can you find a simple exact formula for $\displaystyle\sum_{k=1}^n\frac1k$? – David Nov 20 '15 at 4:01