Infinite Series

Infinite Series
An infinite series is given by the terms of an infinite sequence, added together.
For example, we could take the infinite sequence {2n}={2 ,4 ,6 ,?}
Then the corresponding example of an infinite series would be given by all of these terms added together 2+4+6+8+?
So we would have ?_(n=1)^??2n=2+4+6+8+?
If ??a_n is a positive series,then either ??a_n converges to a positive number,
or??a_n diverges to infinity.
Theorem: The nth-term test for divergence.If lim?(n??)??a_n ??0 ,or if lim?(n??)??a_n ? not exist
then ??a_n diverges.
For example the series? ?_(n=1)^??n^2 diverges because lim?(n??)??n^2 ?=??0
?_(n=1)^??(n+2)/n diverges because lim?(n??)??(n+2)/n?=1?0
?_(n=1)^??(-1)^(n+1) diverges because lim?(n??)??(-1)^(n+1) ? dose not exist

Geometric Series
A geometric series is any series that can be written in the form,
?_(n=1)^???ar^(n-1) ?=a+ar+ar^2+ar^3+?+ar^n+?
1. If |r|<1 then the geometric series converges to the sum s_n=a/(1-r)
2. If |r|?1 then the geometric series diverges to ? .
For example the series ?_(n=1)^??( 1 )/2^n =( 1 )/( 2 )+( 1 )/( 4 )+( 1 )/( 8 )+( 1 )/( 16 )+?
converges because r=1/( 2 )<1 and the sum is s_n=(1?2)/(1-(1?2) )=1.
The series ?_(n=1)^??(3/( 2 ))^n diverges to ? because r=3/( 2 )>1
The p-series
If p is a real constant,the series?_(n=1)^??( 1 )/n^p =( 1 )/1^p +( 1 )/2^p +( 1 )/3^p +?
1.converges if p>1 . 2. diverges if p?1 .
For example the series ?_(n=1)^??1/n^2 converges because p=2>1
The series ?_(n=1)^??1/?(n )=?_(n=1)^??1/n^(1?2) diverges to ? because p=1/( 2 )<1

Tests for converges of series
1. Integral Test
Let the function f(x)=a_n (x) be continuous, positive and decreasing then the
series?_(n=1)^??a_n and the integral ?_1^??f(x) dx both converge or both diverge.
Example 1: Determine whether the series converges or diverges.
1. ?_(n=1)^??1/(n^2+1) 2. ?_(n=1)^??n/(n^2+1) 3. ?_(n=1)^??1/(n^2-n-3)
1. ? ?_1^??dx/(x^2+1)=tan^(-1)?x ?|_1^?=?/2-?/4=?/4 converge
2.?_1^??xdx/(x^2+1)=? 1/2 ln?|x^2+1| ?|_1^? =? diverge
3. 1/(x^2-x-2)=A/(x-2)+ B/(x+1) ? x=2 ?A=( 1 )/( 3 ) and x=-1 ?B=-( 1 )/( 3 )
?_1^??dx/(x^2-x-2)=( 1 )/( 3 ) ? ?_1^??(1/(x-2)- 1/(x+1)) dx=( 1 )/( 3 ) ln?|(x-2)/(x+1)| ?|_1^?
=? ( 1 )/( 3 ) ln?|(1-( 2 )/x)/(1+( 1 )/x)| ?|_1^?=( 1 )/( 3 ) ln?2 converge
2. Ratio Test
Let ?_(n=1)^??a_n be a series with non-negative terms and lim?(n??)??a_(n+1)/a_n ?=L .
If 1. L<1 then the series converges
2. L>1 then the series diverges
3. L=1 then this test is inconclusive
Example 2: Determine whether the series converges or diverges.
1. ?_(n=1)^??? n?^3/( 3^n ) 2. ?_(n=1)^??n!/2^n 3. ?_(n=1)^??3/(2n+5) 4.?_(n=1)^??5^n/(n 3^n )
1. lim?(n??)??a_(n+1)/a_n ?=lim?(n??)??? (n+1)?^3/( 3^(n+1) )×( 3^n)/? n?^3 ?=( 1 )/( 3 ) lim?(n??) ((n+1)/n)^3
=( 1 )/( 3 ) lim?(n??) (1+( 1 )/n)^3=( 1 )/( 3 )<1 converges
2. lim?(n??)??a_(n+1)/a_n ?= lim?(n??) (n+1)!/2^(n+1) ×(2^n )/n!
=lim?(n??) ((n+1)×n!)/(2^n×2)×(2^n )/n!=lim?(n??) (n+1)/2=?>1 diverges

3. lim?(n??)??a_(n+1)/a_n ?=lim?(n??)??3/(2(n+1)+5)?×( 2n+5 )/( 3 )
=lim?(n??)??(2n+5)/(2n+7)?=1. The test is inconclusive .
So we try another test ? ?_1^??3dx/(2x+5)=( 3 )/( 2 ) ln?|2x+5| ?|_1^?=? diverges
4. lim?(n??)??a_(n+1)/a_n ?=lim?(n??)??5^(n+1)/((n+1) 3^(n+1) )×(n 3^n)/5^n ?
=lim?(n??)??5n/(3(n+1) )?=( 5 )/( 3 )>1 diverges
3. Root Test
Let ?_(n=1)^??a_n be a series with non-negative terms and lim?(n??)??(n&a_n )=L .
If 1. L<1 then the series converges
2. L>1 then the series diverges
3. L=1 then this test is inconclusive
Example 3: Determine whether the series converges or diverges.
1. ?_(n=1)^??((3n+2)/( 4n+1))^n 2. ?_(n=1)^??(1/( n ))^n 3. ?_(n=1)^??2^n/n^2 4. ?_(n=1)^??((n-1)/( n ))^n
1. lim?(n??)??(n&a_n )=lim?(n??)??(3n+2)/( 4n+1)?=( 3 )/( 4 )<1 converges

2. lim?(n??)??(n&a_n )= lim?(n??) 1/( n )=0<1 converges
3. lim?(n??)??(n&a_n )=lim?(n??)??2/?(n&n^2 )?=2/(lim?(n??) ?(n&n))=( 2 )/( 1 )=2 >1 diverges
4. lim?(n??)??(n&a_n )=lim?(n??) (n-1)/( n )=1.The test is inconclusive .
So we try another test lim?(n??) ((n-1)/( n ))^n=lim?(n??) (1+(-1)/( n ))^n=e^(-1)?0 diverges

Find the sum of the following series:
1. ?_(n=1)^??( 2^n )/? 5?^n 2.?_(n=1)^??( 5^n )/? 3?^2n 3.?_(n=1)^??( 4 )/? 3?^(n+1)
Determine whether the series converges or diverges. Give reasons for your answers.
4. ?_(n=1)^??(n/( 2n-1 ))^n
5. ?_(n=1)^??( 1 )/(? n?^2 ?n)
6. ?_(n=1)^??(n-1)!/( n !)
7. ?_(n=1)^??( ?11?^n )/? 3?^2n
8. ?_(n=1)^???ne^(-n^2 ) ?
9.?_(n=1)^??? e^(-?n)/?n?
10. ?_(n=1)^??((3n+4)/( 2n ))^(-n)
11. ?_(n=1)^??1/( (3n+1)^3 )
12. ?_(n=1)^????(n+3)!/(2^n n!)