nathrontheking
28.08.2019 •
Mathematics
Consider the series ∑n=1[infinity]ln(n/(n+5)) determine whether the series converges, and if it converges, determine its value. converges (y/n):
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Ответ:
It will diverge. Consider the Nth partial sum,
= (ln1 - ln6) + (ln2 - ln7) + (ln3 - ln8) + (ln4 - ln9) + (ln5 - ln10)
+ (ln6 - ln11) + (ln7 - ln12) + ...
+ (ln(N-5) - lnN) + (ln(N-4) - ln(N+1)) + (ln(N-3) - ln(N+2)) + (ln(N-2) - ln(N+3)) + (ln(N-1) - ln(N+4))
+ (lnN - ln(N+5))
Cancellation among terms leaves us with
= ln1 + ln2 + ln3 + ln4 + ln5 - ln(N+1) - ln(N+2) - ln(N+3) - ln(N+4) - ln(N+5)
We can write this as
Then as , the argument of the logarithm is 0, so the logarithm itself approaches -infinity. Then the limit of the series is non-zero, so it must diverge.
Ответ: