# BSc II Math B Method CH 08

BSc Mathematics chapter 08 “ Infinite Series “.

Introduction :
Infinite series are of great importance in both pure and applied mathematics. Infinite series has significant role in physics and engineering. Many functions can be represented by infinite series.
In this we will discuss some important MCQs taken from topics which are given below.

1) Firstly, we have included some MCQs from topic “ Sequence “.
There are two types of sequence.
a) Finite sequence
b) Infinite sequence.
A sequence is said to be converges if its nth term approaches to definite number.
A sequence that does not have a definite number and its limit is said to be diverge.
Some important theorem of Sequence :
a) A convergent sequence is bounded
b) An unbounded sequence is divergent
c) A bounded monoatomic sequence is convergent.

2) The main and important part of this quiz is Infinite series.
Series may be convergent or divergent.
To check whether series is convergent or divergent, we apply some test.
These test are the basis of infinite series.

a) Divergent test.
If Limit of nth term is not equal to zero, then the nth term series is divergent otherwise test fail.

b) Geometric series test.
If the common ration “r” equal and greater than 1 than series is divergent.
If the common ration “r” less than  1 then series is convergent.

c) Power series test. ( P-series test)
In this test we consider power of “n” in given term like 1/n.
If Power of “n” is less than and equal to 1 than series is divergent.
If power of “n” is greater than one than series is convergent.

d) Basic comparison test.
i) If a(n) and b(n) are positive term with a(n) less than or equal to b(n).
In this case
If b(n) series convergent than a(n) also.
If b(n) diverges, then test fail.
ii) If a(n) and b(n) are positive term with a(n) greater than or equal to b(n).
In this case,
If b(n) divergent then a(n) also.
If b(n) converges, then test fail.

e) Limit Comparison test.
If a(n) and b(n) are positive term
i) If limit a(n)/b(n) = is not equal to zero and infinity.
Then both series behave alike.
ii) If a(n)/b(n) = limit = 0
In this case,
If b(n) convergent, a(n) also.
If b(n) diverges, then test fail.
iii) If a(n)/b(n) = limit = infinity.
If b(n) diverges, then a(n) also
If b(n) converges, then test fail.

f) The integral test. (Named from Famous Scientist Cauchy)
Let a(n) be a positive term series. If function is continuous and non-increasing then we can apply integral test.
If after integration answer is infinity, our series is divergent.
Otherwise series is convergent.

g) The root test. ( d’ Alembert’s test) A French mathematician.
If a(n) be a positive term series test and suppose that lim a(n+1)/a(n) = limit
i) if limit less than 1,  Given series is convergent.
ii) If limit is greater than 1, Given series is divergent.
iii) If limit is equal to 1, then test fail.(We have to apply another test).

h) The root test. ( Cauchy Root test)
if a(n) be a series of positive term and suppose the lim [a(n)]1/n = limit
i) if limit less than 1,  Given series is convergent.
ii) If limit is greater than 1, Given series is divergent.
iii) If limit is equal to 1, then test fail.(We have to apply another test).

i) Alternating series test.
The alternating series ( series with alternating positive and negative sign )
Converges, if the following two conditions hold.
i) nth term series is non increasing sequence.
ii) nth term limit is equal to zero.

Absolute and conditional convergence :
i) A series a(n) is said to be converge absolutely if the series |a(n)| converges.
ii) A series a(n)is said to converges conditionally, if  nth term series converges and |a(n)| diverges.
These are the important test of infinite series.
By applying these test, we can check whether series is convergent or divergent.