Thread: tips on convergence test/partial nums

1. tips on convergence test/partial nums

I know all the series tests for convergences for my calc 2 class. However, there is another technique that involves finding Partial sums to find a general Sn. so, if we find the limit of Sn, it will dictate whether the series diverge or converges

for example, ln(1+1/n) involves this technique...

my question is what is the biggest hint that a series would require this method or recognizing to use this technique (for any exam)

im afraid i would just waste time on using these series test...

2. Re: tips on convergence test/partial nums

It looks to me like your "other technique" is really just the definition of the sum of an infinite series- that it is the limit of the sequence of partial sums.

When you say "for example, ln(1+1/n) involves this technique..." are you referring to $\displaystyle \sum ln(1+ 1/n)$? The partial sums are ln(2)+ ln(3/2)+ ln(4/3)+ ...+ ln(1+ 1/n)= ln(2(3/2)(4/3)...(n/(n-1))(n+1)/n))= ln(n+1) which does not converge.

I think that, generally, you will find the various "tests" of converges (ratio test, power test, etc.) simpler that directly using the definition of series convergence and would use that only if the other tests did not work.

3. Re: tips on convergence test/partial nums

ahh. i remember now. my professor went over this the first day we started learning about sequences or something.

anyway, isnt this old 'method' used with like the telescoping series?

4. Re: tips on convergence test/partial nums

how would i find the convergence of this series ? it seems like i would need to find Sn here and evaluate the limit.

1*3*5...(2n-1) / 2*5*8...(3n-1)

n>infinity