Which series converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. sum_{n=0}^infty(ln(4e^n-1)-ln(2e^n+1))

floymdiT

floymdiT

Answered question

2021-02-06

Which series converge, and which diverge?If a series converges, find its sum. 
n=0(ln(4en1)ln(2en+1))

Answer & Explanation

i1ziZ

i1ziZ

Skilled2021-02-07Added 92 answers

For the infinite series, locate the nth term.
Infinite series: n=0(ln(4en1)ln(2en+1)) 
So n-th term is a_n. 
an=ln(4en1)ln(2en+1) 
To determine whether the series converges or diverges, use the divergence test for the nth term.
limnan=limn[ln(4en1)ln(2en+1)] 
Use the properties for logarithm to simplify. 
limnan=limn[ln(4en1)(2en+1)] 
=limn[ln(4en1)en(2en+1)en] 
=limn[ln41en2+1en] 
Further simplify. limnan=ln(41e)(2+1e) 
=ln(41)(2+1) 
=ln(40)(2+0) 
=ln20 
So, limnan0 
Hence, the series n=0(ln(4en1)ln(2en+1)) diverges.

Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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