# Math 2     Examples of Series      15 Nov 96
# 
# 
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# First example:
> a1 := n -> (n+1)/(3*n+1);

                                           n + 1
                               a1 := n -> -------
                                          3 n + 1
# Partial sums:
> Sn1 := n -> sum(a1(k),k=1..n);

                                           n
                                         -----
                                          \
                             Sn1 := n ->   )   a1(k)
                                          /
                                         -----
                                         k = 1
# Plot a1 and Sn1:
> A1 := [seq([n,a1(n)],n=1..200)]:   B1 := [seq([n,Sn1(n)],n=1..20)]:
> plot(A1);
> plot(B1,style=point);
# We get infinity if we try to find the limit of the partial sums:
> Sn1(infinity);

                                    infinity
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# Second example:
> a2 := n -> (2*n**2+3)/sqrt(n**7-2*n**3+2);

                                             2
                                          2 n  + 3
                         a2 := n -> -------------------
                                          7      3
                                    sqrt(n  - 2 n  + 2)
> Sn2 := n -> evalf(sum(a2(k),k=1..n));

                                             n
                                           -----
                                            \
                         Sn2 := n -> evalf(  )   a2(k))
                                            /
                                           -----
                                           k = 1
# Plot a2 and Sn2:
> A2 := [seq([n,a2(n)],n=1..100)]: B2:= [seq([n,Sn2(n)],n=1..100)]:
> plot(A2,style=point,color=red);
> plot(B2,style=point);
# Find the limit of the partial sums:
> Sn2(infinity);

                                   8.669926180
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# Third example:
> a3 := n -> 1/(n*ln(n));

                                             1
                               a3 := n -> -------
                                          n ln(n)
# Partial sums:
> Sn3 := n -> evalf(sum(a3(k),k=2..n));

                                             n
                                           -----
                                            \
                         Sn3 := n -> evalf(  )   a3(k))
                                            /
                                           -----
                                           k = 2
# Plot a3 and Sn3:
> A3 := [seq([n,a3(n)],n=2..200)]: B3 := [seq([n,Sn3(n)],n=2..200)]:
> plot(A3,color=red);
> plot(B3);
# MAPLE is unable to find the limit of the partial sums:
> Sn3(infinity);

                                FAIL(3.395407688)
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# Fourth example:
> a4 := n -> cos(n*Pi)/sqrt(n);

                                         cos(n Pi)
                              a4 := n -> ---------
                                          sqrt(n)
# Partial sums:
> Sn4 := n -> evalf(sum(a4(k),k=1..n));

                                             n
                                           -----
                                            \
                         Sn4 := n -> evalf(  )   a4(k))
                                            /
                                           -----
                                           k = 1
# Plot a4 and Sn4:
> A4 := [seq([n,a4(n)],n=1..100)]: B4 := [seq([n,Sn4(n)],n=1..100)]:
> plot({A4,B4},style=point);
# Find the limit of the partial sums:
> Sn4(infinity);

                                  -.6048986434
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# Fifth example:
> a5 := n -> 2^(2*n)/n!;

                                            (2 n)
                                           2
                                a5 := n -> ------
                                             n!
# Partial sums:
> Sn5 := n -> evalf(sum(a5(k),k=1..n));

                                             n
                                           -----
                                            \
                         Sn5 := n -> evalf(  )   a5(k))
                                            /
                                           -----
                                           k = 1
# Calculate the ratios for the ratio test:
> c5 := n -> abs(a5(n+1)/a5(n));

                                           a5(n + 1)
                            c5 := n -> abs(---------)
                                             a5(n)
# Plot a5, Sn5, and also b5:
> A5 := [seq([n,a5(n)],n=1..20)]:  B5 := [seq([n,Sn5(n)],n=1..20)]:
> C5 := [seq([n,c5(n)],n=1..50)]:
> plot(A5,style=point,color=red);
> plot(B5,style=point);
> plot(C5,style=point);
# Find the limit of the partial sums:
> Sn5(infinity);

                                   53.59815003
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# Final example:
> a6 := n -> (-2)^n/((ln(n))^n);

                                                n
                                            (-2)
                                a6 := n -> ------
                                                n
                                           ln(n)
# Partial sums:
> Sn6 := n -> evalf(sum(a6(k),k=2..n));

                                             n
                                           -----
                                            \
                         Sn6 := n -> evalf(  )   a6(k))
                                            /
                                           -----
                                           k = 2
# Let c6 be the nth root of a6, which we use in the root test:
> c6 := n -> -2/ln(n);

                                              2
                               c6 := n -> - -----
                                            ln(n)
# Plot a6, Sn6, and also the root c6:
> A6:=[seq([n,a6(n)],n=2..20)]: B6:=[seq([n,Sn6(n)],n=2..20)]:
> C6 := [seq([n,c6(n)],n=2..500)]:
> plot(A6,style=point,color=red);
> plot(B6,style=point);
> plot(C6,color=yellow);
# Find the limit of the partial sums:
> Sn6(infinity);

                                   4.843496974
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