Match each of the following with the correct statement.
A. The series is absolutely convergent.
C. The series converges, but is not absolutely convergent.
D. The series diverges.
Can someone please help me?
there are numerous ways to test converge of sums of infinite series, however the most common of which is to use the ratio test,
Ratio Test :
Given an infinite sum of the form
S = sum (a(n), n = 0..inf)
then if L < 1, the series converges, L > 1 the series diverges and finally if L = 1 the test fails and nothing can be stated (by the Ratio test) about its convergence,
L = lim (n--> inf) |a(n+1)/a(n)|
As an example I will do one of your questions where
a(n) = (-1)^n * (1/n)*(1/ln(n+3))
then a(n+1) = (-1)^(n+1) * (1/(n+1))*(1/ln(n + 4))
a(n+1)/a(n) = (-1)^(n+1) * (1/(n+1))*(1/ln(n + 4)) / ((-1)^n * (1/n)*(1/ln(n+3)))
|a(n+1)/a(n)| = (n/(n+1))*(ln(n+3)/ln(n+5))
we now need to evaluate the limit,
L = lim (n --> inf) (n/(n+1))*(ln(n+3)/ln(n+5))
note it is easily observed that lim (n--> inf) (n/(n+1)) = 1 then L becomes
L = lim (n --> inf) ln(n+3)/ln(n+5)
Hence employ L'hopitals Rule
L = lim (n--> inf) d/dn (ln(n+3))/ d/dn (ln(n+5))
L = lim (n--> inf) (1/(n+3))/(1/(n+5))
L = lim (n--> inf) (n+5)/(n+3) = 5/3 > 1
Thus the series diverges.
Hope this was informative and I didnt go overboard with the description.
ps - forgot to mention that for something to be absolutely convergent then the sum( |a(n)| , n = 0..inf ) must converge, in order to determine this, simply take the absolute value of the integrand of summation a(n) and apply the Ratio test as prescribed.