Find the fourth term of the sequence: {(-1)^n3^n + 1/2n-1},n = 1, 2, 3,.... Dete

Question

Find the fourth term of the sequence: {(-1)^n3^n + 1/2n-1},n = 1, 2, 3,.... Determine if the following sequence converges or diverges: {2^n/1 +n},n = 1, 2, 3,.... If the sequence converges, find its limit 1.2,3. 3. Determine if the following sequence converges or diverges: {(n + 1/n)^n},n = 1, 2, 3,... If the sequence converges, find its limit. Find the sum: Find the For each of the series in 6 through 14. determine convergence or divergence and state the test used. Find the number of terms necessary to approximate the sum of the series with an error of less than 0.001: Find the radius of convergence of the series

Explanation / Answer

((n+1)/n)n

limn->((n+1)/n)n

limn->(1+(1/n))n

let limit =y

y=limn->(1+(1/n))n

apply logaritham on both sides

lny =limn->ln(1+(1/n))n

lny =limn-> nln(1+(1/n))

lny =limn-> (ln(1+(1/n)))/(1/n)

we get 0/0 form, apply l hospitals rule, differentiate numerator and denominator with respect to n

lny =limn-> [(1/(1+(1/n)))(0+(-1/n2))]/(-1/n2)

lny =limn-> [(1/(1+(1/n)))(-1/n2)]/(-1/n2)

lny =limn-> (1/(1+(1/n)))

lny =(1/(1+(0)))

lny =1

y =e

limn->ln(1+(1/n))n=e

sequence converges and limit =e

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