Find the sum of that alternating series given by 1 12 13

Find the sum of that alternating series given by 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 +...

Solution

The given series is known as the alternating harmonic series. In particular, the sum is equal to the natural logarithm of 2.

% our goal
L = log(2)

% we try 5 terms
n = 1 : 5

% define the alternating sequence
seq = (-1).^(n+1) ./ n

% get the sum
s = sum(seq)

The Matlab answer is:

L =     0.6931
n =     1     2     3     4     5
seq =   1.0000   -0.5000    0.3333   -0.2500    0.2000
s =     0.7833

We see that 5 terms are not enough. Let’s try 1000 terms.

% we try 1000 terms
n = 1 : 1000;

% define the alternating sequence
seq = (-1).^(n+1) ./ n;

% get the sum
s = sum(seq)

The new answer is:

s =    0.6926

This is a closer answer, but we’re still far from log 2. More terms are needed for higher accuracy... be aware...