2. Which of the following series converge conditionally?E(-1)- 2+(-1)"3+ (-1)"O 1, 1.31-3-5 --- (2n – 1)+(-13 - 5 - 7 -- (2n + 1)1.3.5--+3' 3.5+..3.5-7E(-1)" sinn+n=1E(-1)" In "n+1cos(2n)!1.3-5 ..- (2n – 1)E(-1)"neEl-1)".Vn(n+2)1E(-1)"n² tann=1

Name: 2. Which of the following series converge conditionally?E(-1)- 2+(-1)
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: 2. Which of the following series converge conditionally?E(-1)- 2+(-1)"3+ (-1)"O 1, 1.31-3-5 --- (2n – 1)+(-13 - 5 - 7 -- (2n + 1)1.3.5--+3' 3.5+..3.5-7E(-1)" sinn+n=1E(-1)" In "n+1cos(2n)!1.3-5 ..- (2n – 1)E(-1)"neEl-1)".Vn(n+2)1E(-1)"n² tann=1

We will answer first 3 questions since we don't answer multiple questions at a time. Please resubmit the other questions specifying which one you would like to be answered.1) The given expression is ∑n=1∞-1n2+-1n3+-1nLet, un=2+-1n3+-1n Now, limn→∞un=limn→∞2+-1n3+-1n→∞ So, limn→∞un≠0Hence, ∑n=1∞-1n2+-1n3+-1n is not convergent that is the given d=series is not convergent.2) The given expression is -13+1.33.5-1.3.53.5.7+...+-1n1.3.5...2n-13.5.7...2n+1+...Let, un=1.3.5...2n-13.5.7...2n+1 limn→∞un=limn→∞1.3.5...2n-13.5.7...2n+1=limn→∞1n.3n.5n...2-1n3n.5n.7n...2+1n=0 Now, un+1un=2n+12n+3<1⇒un+1<un So, un is monotone decreasing. By D'Alemberts' test limn→∞un+1un=limn→∞2n+12n+3=limn→∞2+1n2+3n=1 (not conclusive) By Rabees test limn→∞nunun+1-1=limn→∞n2n+32n+1-1=limn→∞n22n+1=limn→∞22+1n=1 (not conclusive) Applying Morgan and Bertands test limn→∞nunun+1-1-1logn=limn→∞2n2n+3-1logn=limn→∞-32n+3logn=limn→∞-3n2n+3lognn=0<1 So, un is divergent.Hence, ∑n=1∞-1nun that is the given series is conditionally convergent.3) The given series is ∑n=1∞-1nsin1nLet, un=sin1n Now, limn→∞un=limn→∞sin1n=sin0=0 Also, n+1>n⇒1n+1<1n⇒sin1n+1<sin1n⇒un+1<un So, un is monotone decreasing and also ∑n=1∞-1nsin1n is monotone decreasing. Now, un=sin1n=1n-13!1n3+15!1n5-17!1n7+... and vn=1n So, limn→∞unvn=limn→∞1-13!.1n2+15!.1n4-17!.1n6=1 which is a finite positive number. So, -1nsin1n is divergent. Hence, ∑n=1∞-1nsin1n is conditionally convergent.