
Series

Infinite series 
Geometric
series 
Pseries 





Geometric
series 
A
series, whose successive terms differ by a constant multiplier, is called a geometric series
and written as 

If
 x  < 1 ;
the nth
partial sum is 

Thus, the geometric series is convergent if
 x  < 1
and its sum is 

If
 x  > 1
then the geometric series
diverges. 

Example: Show
that the series 

converges. 

Solution:
Given
is the geometric series subsequent terms of which are multiplied by the
factor 1/2. 



Example: Let
prove
that the pure recurring decimal 0.333
. . .
converges to 1/3.

Solution:
Given
decimal can be written as 


Example: Let
calculate the square of the convergent
geometric series


using
the Cauchy product shown above. 
Solution:



= 1 + 2x + 3x^{2}
+ 4x^{3} + · · · + (n + 1)x^{n} + · · · 
the
obtained series 

converges
for 0 < x^{}
<1. 



Pseries 
The
series 

converges
if
fixed constant
p
> 1 and diverges if p
< 1. 

By
grouping terms 


where
the number of terms in parentheses form the sequence 2,
4, 8, ... 2^{r1},
... such that 

therefore,
each value in parentheses is smaller than the corresponding term of the geometric
series 

Thus,
if p
> 1
then q
< 1,
the geometric series converges so that the given series is also
convergent. 
Euler
discovered and revealed sums of the series for
p
=
2m,
so for example 

If
p < 1
then n^{p}
< n or 1/n^{p}
> 1/n, therefore the
terms of the given series are not smaller than the terms of
the divergent harmonic series so, given series diverges. 









Calculus
contents B 



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