MATH 166 SCW Lecture 19 Work for Examples 3, 4, and 5
MATH 166 SCW Lecture 19 Work for Examples 3, 4, and 5
Example 3
Example 3
Find some partial sums of the series , for increasing values of n:
∞
∑
k=1
1
1+
2
k
TableFormTable10^n,Sum,{k,1,10^n},{n,1,7},TableHeadings->None,"n","="
1.
1+k^2
S
n
n
∑
k=1
1
1+
2
k
Out[]//TableForm=
n | S n n ∑ k=1 1 1+ 2 k |
10 | 0.981793 |
100 | 1.06672 |
1000 | 1.07567 |
10000 | 1.07657 |
100000 | 1.07666 |
1000000 | 1.07667 |
10000000 | 1.07667+0. |
Check if Mathematica can find the actual sum S for this series, i.e .
S=
∞
∑
k=1
1
1+
2
k
Sum,{k,1,∞}
1
1+k^2
Out[]=
1
2
N[%]
Out[]=
1.07667
Example 4
Example 4
Sum[1/k^5,{k,1,23}]
Out[]=
1106136860583472631494090651392248539192679773
1066745227156578837062475058249339864272076800
N[%]
Out[]=
1.03693
Example 5
Example 5
TableFormTablen,DecimalFormSum,{k,1,n}+,7,DecimalFormSum,{k,1,n}+,7,DecimalForm12*Sum,{k,1,n}++Sum,{k,1,n}+,7,DecimalForm12*Sum,{k,1,n}+-Sum,{k,1,n}+,7,{n,1,15},TableHeadings->{None,{n,"Lower Est.","Upper Est.","Average Est.","Error Bound"}}
1.
5
k
1
4
4
(n+1)
1.
5
k
1
4
4
n
1.
5
k
1
4
4
n
1.
5
k
1
4
4
(n+1)
1.
5
k
1
4
4
n
1.
5
k
1
4
4
(n+1)
Out[]//TableForm=
n | Lower Est. | Upper Est. | Average Est. | Error Bound |
1 | 1.015625 | 1.25 | 1.132813 | 0.1171875 |
2 | 1.034336 | 1.046875 | 1.040606 | 0.00626929 |
3 | 1.036342 | 1.038452 | 1.037397 | 0.001054929 |
4 | 1.036742 | 1.037318 | 1.03703 | 0.0002882813 |
5 | 1.036855 | 1.037062 | 1.036958 | 0.0001035494 |
6 | 1.036895 | 1.036983 | 1.036939 | 0.00004438898 |
7 | 1.036911 | 1.036954 | 1.036932 | 0.00002154406 |
8 | 1.036919 | 1.036941 | 1.03693 | 0.0000114656 |
9 | 1.036922 | 1.036935 | 1.036929 | 0.000006551974 |
10 | 1.036924 | 1.036932 | 1.036928 | 0.000003962332 |
11 | 1.036926 | 1.036931 | 1.036928 | 0.000002509505 |
12 | 1.036926 | 1.03693 | 1.036928 | 0.000001651566 |
13 | 1.036927 | 1.036929 | 1.036928 | 0.000001122745 |
14 | 1.036927 | 1.036929 | 1.036928 | 0.0000007847168 |
15 | 1.036927 | 1.036928 | 1.036928 | 0.0000005617872 |