Series
Mr. V. V. Chandavale
Asst Prof
Department of Mathematics
Raje Ramrao Mahavidyalaya, Jath
Infinite Series
Given an infinite sequence , then
is called an infinite series.
Note
is the infinite sequence
is an infinite series.
Consider
Note that is a sequence of
numbers called a sequence of partial sums.
Definition
For an infinite series , the partial sum is
given by
If the sequence of partial sums converges to S,
the series converges.
The limit S is the sum of the series
If the sequence diverges, then the series
diverges.
Example 1: Find the first five sequence of partial
sum terms of the series
Find a formula that describes the sequence of
partial sums and determine whether the
sequence converges or diverges.
Solution:
Example 2: Find the first five sequence of partial
sum terms of the series
Find a formula that describes the sequence of
partial sums and determine whether the
sequence converges or diverges.
Solution:
Geometric Series
A geometric series is given by
with ratio r.
Notes
. When , the sum of the series (the value the series converges to) is
If , then the geometric series diverges.
2. The value a is the first term of the series.
3. The ratio r is the factor you multiply the previous term by to get the next one. That is,
Example 3: Determine whether the series
is convergent or divergent. If convergent,
find its sum.
Solution:
Example 4: Determine whether the series
is convergent or divergent. If
convergent, find its sum.
Solution:
Example 5: Determine whether the series
is convergent or divergent. If convergent,
find its sum.
Solution:
Properties of Series (p. ??)
Example 6: Determine whether the series
is convergent or divergent.
If convergent, find its sum.
Solution:
Applications of Geometric Series
Example 7: Express as a
ratio of integers.
Solution: (In typewritten notes)
Tests For Non-Geometric Series
Most series are not geometric – that is, there is
not a ratio r that you multiply each term to get to
the next term. We will be looking at other ways to
determine the convergence and divergence of
series in upcoming sections.
Some other ways to test series
does not converge to 0, then the series
diverges.
Note: This is only a test for divergence – if the sequence converges to 0 does not necessarily mean the series converges.
2. Examine the partial sums to determine convergence or divergence (Examples 1 and 2
of this section).
3. Techniques discussed in upcoming sections.
Example 8: Demonstrate why the series
is not geometric. Then
analyze whether the series is convergent or
divergent.
Solution:
Example 9: Analyze whether the series
is convergent or divergent.
Solution: