Convergence of series minus logarithmim trying to solve this problem since two, three days.. Is...
Convergence of series minus logarithm
im trying to solve this problem since two, three days.. Is there someone who can help me to solve this problem step by step. I really want to understand & solve this!
The sum looks like the harmonic series.
My thoughts were to compare this sum with an integral, the lower and the upper riemann-sum, to get an inequation.
Answer & Explanation
Comparing sum and integral, we get
By the Mean Value Theorem and since is decreasing, for some
Therefore, the red difference in is decreasing in since
and by , each blue term in is negative.
So, says that the red difference is decreasing and bounded below by . Thus, the limit of the red difference exists and is at least
Furthermore, also says that for each , the red difference is at most
The function is continuous non negative decreasing on hence the sequence
is convergent, in fact
Can you take it from here?