series of non negative terms

1. ADVANCED CALCULUS- 11
Submitted to :- Shilpa Mam
Presented by :- Harjas kaur
Class :- B.sc ( non med.)
Roll no. :- 2034002

2. CHAPTER-
INFINITE SERIES
PAGE NO;-99

3. CONCEPTS TO BE COVERED:-
 Series of non negative terms .
 P – test
 Comparison test
 Cauchy’s integral test
 Ratio test
 Kummer’s test

4. First of all we will know about :- * what are series?
*what are infinite series?
Series:- A series is a sum of a sequence of terms. That is
a series is a list of numbers with addition operators
between them.
Example :- { 1,4,9,16,25,36………….}

5. Infinite series :- Let {𝑎𝑛} be a sequence of real
numbers. The expression 𝑎1 + 𝑎2 + 𝑎3 +………𝑎𝑛 is
called an infinite series and is denoted by 𝑛=1
∞
𝑎𝑛 or by
∑𝑎𝑛 and 𝑎𝑛 is called the nth term of series.
Example :- 𝒔𝒏 = 𝒂𝟏 + 𝒂𝟐 + 𝒂𝟑 + … . . +𝒂𝒏
= 𝟏
𝒏
𝒂𝒏

6. Series of Non-Negative terms
When 𝑎𝑛 is a non negative real number for every n, the
sequence 𝑆𝑛 of partial sum is non decreasing . It follows
that a series 𝑎𝑛 with non negative terms converges if and
only if the sequence 𝑆𝑛 of partial sum is bounded.

7. WHY ARE CONDUCTING
THESE TESTS?
 So far we have proved that a series converges by actually finding its sum.
 But for most convergent series the exact sum is difficult to find .
 This is true for cases like ∑1/𝑛2 or ∑1/𝑛3 . Yet in such cases it may be
sufficient to know at least the series converges .
 We shall formulate tests for determining the convergence and
divergence of the series.
 For present we shall restrict our attention to non negative series, that is ,
to series whose terms are non negative.

8. P – TEST

9. Any series of the form 1
∞
1/𝑛𝑝
where p is a real
number.
p series converges if and only if p >1.
1. If p =1 then 1
∞
1/𝑛𝑝 is the divergent harmonic
series.
2. If p> 1, then the series converges .
3. If p<1, then the series diverges.
Examples :-pg 128-pg 136.

10. THE COMPARISON TEST

11.  Let {an} be a sequence of nonnegative real numbers, let {𝑏𝑛} be a
sequence of positive real numbers, and assume that lim
𝑛→∞
𝑎𝑛
𝑏𝑛
= L .
There are three cases:
 1. If L is a finite positive real number, then the series ∑𝑎𝑛and
∑𝑏𝑛converge or diverge simultaneously.
 2. If L = 0, then the convergence of ∑𝑏𝑛 implies the convergence
of ∑𝑎𝑛, or the divergence of ∑𝑎𝑛 implies the divergence of ∑𝑏𝑛
 3. If L = +∞, then the convergence of ∑𝑎𝑛 implies the
convergence of ∑𝑏𝑛, or the divergence of 𝑏𝑛 implies the divergence
of ∑𝑎𝑛.

12. CAUCHY’S INTEGRAL
TEST

13. It is a test that compares the non negative series with an
improper integral.
Let f(1) + f(2)+………..+f(n)+…….be a non negative
series , and let f be a continuous , decreasing function
defined [1,∞). Then the series ∑ f(n) converges or diverges
according as the integral 1 to ∞ f(x)dx is finite or infinite.

14. THE RATIO TEST

15. Let ∑Un be a non negative series . Assume Un is not
equal to 0 for all n and that lim
𝑛→∞
Un+1/Un=L
a. If 0 ≤ L < 1, then ∑ Un diverges.
b. If L >1 , then ∑ Un converges.
If L =1 , then from this test alone we cannot draw
any conclusion about the convergence or divergence of ∑
Un.

16. KUMMER’S –TEST

17. Given a series of positive terms 𝑢1 and a sequence of finite
positive constants 𝑎1. Let 𝜌 = lim
𝑛→∞
( 𝑎𝑛
𝑈𝑛
𝑈𝑛+1
- 𝑎𝑛+1).
1. If 𝜌>0, the series converges.
2. If 𝜌 < 0 and the series 𝑛=1
∞ 1
𝑎𝑛 diverges, the series
diverges.
3. If 𝜌 = 0,the series may converge or diverge.

18. Example of p – test
Ques :-discuss the convergence and divergence of the series
𝑛2+1-n.
Sol:- the given series is 𝑎𝑛
𝑎𝑛= 𝑛2+ 1 –n
𝑛2+1-n × 𝑛2 +1+n / 𝑛2+1+n
=𝑛2
+ 1 -𝑛2
/ 𝑛2+ 1 + n

19. let 𝑏𝑛 =1/n
=𝑎𝑛/𝑏𝑛=1/ 𝑛2 + 1 + n ×n/1
=n/ 𝑛2 + 1 + 𝑛
=1/ 1 + 1/𝑛2
+ 1
lim
𝑛→∞
𝑎𝑛/𝑏𝑛
= 1/ 1 + 0 + 1
=1/1+1
=1/2
Which is finite and non zero value
∴ 𝑎𝑛 𝑎𝑛𝑑 𝑏𝑛 will converge and diverge together.

20. THANK YOU