MATH 132 -- Calculus for Physical and Life Sciences 2

April 8, 2005

The alternating harmonic series is sum((-1)^n/n,n = 1 .. infinity)   .   Question:  Does

this converge?   

Here are some of the partial sums:   sum((-1)^n/n,n = 1 .. N)

N Partial*Sums
10 .6456349206
50 .6832471606
100 .6881721793
500 .6921481806
1000 .6926474306
5000 .6930471906
10000 .6930971831


Here's another way to visualize what the partial sums are doing.  The plot

shows the partial sum values for   N = 1 .. 50.  Note that they seem to be

oscillating back and forth around roughly .69 ...

>    SList:=[seq([N,evalf(add((-1)^(n+1)/n,n=1..N))],N=1..50)]:

>    with(plots):

>    A:=plot(SList,style=point,color=red):

>    B:=plot(SList,color=blue):

>    display(A,B);

[Maple Plot]

So it looks like this series does converge to a number approximately  .6931 ... Does this  look familiar?   

>    evalf(ln(2));

.6931471806

>