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cases where the sequence may approach a limit, while the series is divergent.
 1 
The classical example is that of the sequence  ; this sequence approaches
 n
the limit zero, while the corresponding series is divergent.
In any numerical calculation, we cannot perform the operation of adding
an infinite number of terms. We can only add a finite number of terms. The
infinite sum of a convergent series is the limit of the partial sums S .
N
You will study in your calculus course the different tests for checking the
convergence of a series. We summarize below the most useful of these tests.
• The Ratio Test, which is very useful for series with terms that
contain factorials and/or n power of a constant, states that:
th

∞
+1
for a > 0, the series ∑ a is convergent if lim   a n   < 1
n n n→∞  a 
n=1 n
∞
• The Root Test stipulates that for a > 0, the series ∑ a n is conver-
n
gent if n=1

lim( )a n 1 /n < 1
n→∞
• For an alternating series, the series is convergent if it satisfies the
conditions that

lim a = 0 and a n+1 < a n
n
n→∞
Now look at the numerical routines for evaluating the limit of the partial
sums when they exist.

Example 4.4 N n
N ∑
Compute the sum of the geometrical series S =   1 .
  2
n=1
Solution: Edit and execute the following script M-file:

for N=1:20
n=N:-1:1;
fn=(1/2).^n;
Sn(N)=sum(fn);
end
NN=1:20;
plot(NN,Sn)

© 2001 by CRC Press LLC