Ok, note that is a decreasing sequence, and , thus by Leibniz Test converges
Alternating series test - Wikipedia, the free encyclopedia
You can do the same for the second one, or see that converges, thus must be convergent
Just to clarify on PaulRS's completely correct statement
If converges this means that it converges absolutely
and if a series converges absolutely it converges conditionally(with the
But the converse is not necassarily true
I will use these two as examples
For the first one we first test for absolute convergence
So we check if converges
To this we notice that all the terms are positive and
So the integral test applies
So we set up the integral test
and so since the integral diverges the series diverges. So we know that this series is not absolutely convergent
but now we check for conditional convergence. To do this we use the alternating series test which states that if a series of the form and which means that if you can pick any number and after than number is monotonically decreasing ( )
If this applies then we can use the altnerating series test which states
that if the series is convergent
if the series is divergent
So since our series
Meets this criteria (e.g.
the alternating series test applies
So we check to see if
since direct substitution yields no indeterminate forms and its value is 0 we conclude the limit is 0
Therefore this series converges by the alternating series test
That showed that the converse of "if a series converges absolutely it converges conditionally" is untrue
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Now for your other series
we check for absolute convergence
so first we find
So either seeing that this is a convergent p-series or seeing that
so since the integral converges the series converges.
Now since we have shown that the series absolutely converges it MUST by definition converege conditionally
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Remember to follow instructions though. Generally when they ask you to find the convergence they mean conditional not absolute