determine whether the series 1 + 1/2^5 + 1/3^5 + 1/4^5 +...is convergent or divergent?
How do I tell the difference?
do the ratio test:
ratio (An+1)/An=
the series is SUM (1/n)^5 for n=1>inf
ratio [(1/n+1)/ (1/n)]^5=(n/(n+1))^5 which is less than 1, so the series converges. This means, if you add all the terms, the sum will be a finite q
do the ratio test:
ratio (An+1)/An=
the series is SUM (1/n)^5 for n=1>inf
ratio [(1/n+1)/ (1/n)]^5=(n/(n+1))^5 which is less than 1, so the series converges. This means, if you add all the terms, the sum will be a finite quanity. If the ratio had been 1 or greater, the series would have been divergent.
http://abacus.bates.edu/acad/acad_support/msw/convergence_tests.pdf
To determine whether the series 1 + 1/2^5 + 1/3^5 + 1/4^5 + ... is convergent or divergent, you can use the concept of convergence in mathematics.
There are several tests you can use to determine the convergence or divergence of a series, but one commonly used test is the p-series test. The p-series test states that if a series can be written in the form Σ(1/n^p), where p is a positive constant, then the series is convergent if p > 1 and divergent if p ≤ 1.
In the given series, we have Σ(1/n^5), where n represents the increasing term numbers. In this case, p = 5, so we can apply the p-series test. Since p > 1 (in this case, p = 5), the series 1 + 1/2^5 + 1/3^5 + 1/4^5 + ... is convergent.
Therefore, the series converges.