State whether the series converges or diverges. And state which test was used. T

Question

State whether the series converges or diverges. And state which test was used.
The sum of where n=1 to Infiniti n^2+3n-4/5n^2-6

Also the sum of where n=0 to Infiniti (n+1)!/25^n

And the sum of where n=2 to Infiniti ln n/n

Explanation / Answer

now if we see that its nth terms in not tending to ZERO then the overall sum can be infinite : Tn = 1/5 as n-> infinity : hence sum leads to infinity hence diverging !! B) use ration test : |Tn+1/Tn|: ratio of n+1 term to nth term : (n+2)!/25^(n+1) * 25^n/(n+1)! : (n+2)/25 as n tends to infinity ratio tends to infinity : this implies terms keep on greater than the previous terms hence sum diverges 1!! c) again use ratio test : ln(n+1)/(n+1) * (n)/ln[n] : as n tends to inifnity : to get the limit use L hospital rule L (n/n+1) as ntends to inifnity : 1 as ratio comes out o be 1 , hence it is of divergin nature !!! if ratio had been less than 1 , then only convergent !!!

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