Convergence of series minus logarithm im trying to solve this problem since two, three days..
Makenna Booker
Answered question
2022-07-15
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
iljovskint
Beginner2022-07-16Added 18 answers
Comparing sum and integral, we get
Thus,
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
Jaxon Hamilton
Beginner2022-07-17Added 3 answers
The function is continuous non negative decreasing on hence the sequence