Sequences and Series Name:

restart; a:=array(1..100); for n from 1 to 100 do a[n]:=1/n:od: plot({seq([i,a[i]],i=1..100)},style=point); We can observe that the terms of the sequence approach 0. The limit command can also be used. limit(1/i,i=infinity);

restart; Choose a value for i. i:=300: a:=array(1..i);s:=array(1..i); for n from 1 to i do a[n]:=1/n:od: for j from 1 to i do s[j]:=sum(a[n1],n1=1..j):od: plot({seq([j,s[j]],j=1..i)},style=point); The sequence of partial sums is not approaching a limit. This can be seen by evaluating some of them. evalf(s[100]);evalf(s[150]);evalf(s[200]);evalf(s[250]);evalf(s[300]); This series has been proven to diverge by other methods. The above is not a proof, but can serve to illustrate the idea of divergence.

restart; a:=array(1..100); for n from 1 to 100 do a[n]:=1/n^2:od: plot({seq([i,a[i]],i=1..100)},view=[0..100,0..0.3],style=point); We can observe that the terms of the sequence approach 0. The limit command can also be used. limit(1/i,i=infinity); The limit of the nth term is 0, so the sequence converges and the series could converge. restart; Choose a value for i. i:=300: a:=array(1..i);s:=array(1..i); for n from 1 to i do a[n]:=1/n^2:od: for j from 1 to i do s[j]:=sum(a[n1],n1=1..j):od: plot({seq([j,s[j]],j=1..i)},style=point); Compare this graph with the one in example 2 and observe that it approaches a limit much more quickly and can be shown to converge. Find some partial sums to illustrate this fact. evalf(s[50]);evalf(s[100]);evalf(s[150]);evalf(s[200]);evalf(s[250]); If a series converges, Maple can often find the sum n:='n':#This command initializes n sum(1/n^2,n=1..infinity);

The Ratio Test

For a series 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 , if 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 , then the series converges if L < 1, diverges if L > 1, and can do either if L = 1.

Example 1: Determine if 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 converges. If it does, find the sum. restart; Limit((exp(n+1)/(n+1)!)/(exp(n)/n!),n=infinity); value(%); Since 0 < 1, the series converges. Sum(exp(n)/n!,n=0..infinity); evalf(%); Example 2: Determine if 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 converges. If it does, find the sum. restart; Limit(((n+1)^3/exp(n+1))/(n^3/exp(n)),n=infinity); value(%); The limit is less than 1, therefore the series converges. Sum(n^3/exp(n),n=0..infinity); evalf(value(%)); Example 3: Determine if the series 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 converges. If it does, find the sum. restart; Limit((2^(n+1)/(n+1)^4)/(2^n/n^4),n=infinity); value(%); The limit is greater than 1, therefore the series diverges.

Exercise 3: Use the ratio test to determine if the series 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 converges. If it does, find the sum. restart; Exercise 4: Use the ratio test to determine if the series 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 converges. If it converges, find the sum. restart;

The ratio test can also be used to find the interval of convergence of a power series. Example 1: Find the interval of convergence of the series 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 . restart; f:=(n,x)->(x-2)^n; a:=limit(f(n+1,x)/f(n,x),n=infinity); solve(abs(a)<1); The endpoints of the interval must now be tested. Test_1:=Sum(f(n,1),n=0..infinity); value(%); The series above diverges.. Test_2:=Sum(f(n,3),n=0..infinity); value(%); Therefore, neither enedpoint is included. The interval of convergence is 1 < x < 3. Example 2: Find the interval of convergence of the series 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 . restart; f:=(n,x)->x^n/(n*4^n); a:=limit(f(n+1,x)/f(n,x),n=infinity); solve(abs(a)<1); Test_1:=Sum(f(n,-4),n=1..infinity); value(%); Test_2:Sum(f(n,4),n=1..infinity); value(%); Thus, the interval of convergence is \342\200\2234 <= x < 4.

Exercise 5: Find the interval of convergence of the series 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 . restart; Exercise 6: Determine the interval of convergence of the series 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 . restart;