Problem

2) Find the interval of convergence if
n=1(1)n(x1)n3n4n(n+1)

Answer

Expert–verified
Hide Steps
Answer

The interval of convergence is (13,73)

Steps

Step 1 :Rewrite the series as n=1an, where an=(1)n(x1)n3n4n(n+1)

Step 2 :Apply the Ratio Test: limn|an+1an|=limn|(1)n+1(x1)n+13n+14n+1(n+2)(1)n(x1)n3n4n(n+1)|

Step 3 :Simplify the expression: limn|(x1)34n+1n+2|

Step 4 :Take the limit as n approaches infinity: |3(x1)4|

Step 5 :For the series to converge, the result of the Ratio Test must be less than 1: |3(x1)4|<1

Step 6 :Solve the inequality: 1<3(x1)4<1

Step 7 :Multiply all sides by 4: 4<3(x1)<4

Step 8 :Distribute the 3: 4<3x3<4

Step 9 :Add 3 to all sides: 1<3x<7

Step 10 :Divide all sides by 3: 13<x<73

Step 11 :The interval of convergence is (13,73)

link_gpt