Question bank (unit-2)
Question-1: : Show that
Sol. For ϵ > 0 , we need to find an integer N such that ,
| 
For all n≥N , since |i|=1 we need 1/n< ϵ , so that we choose any N>1 / ϵ>0 , for all n≥N , we find that
Here |i| = 1 , then
Question-2: show that
= 1
Sol.
Let Sn = 
- 1 for all n >0 , here Sn ≥ 0 for all positive n , therefore
For all n>0 ,
for all n > 1 
From Sandwich theorem, we get,
Question-3: check whether the sequence is bounded monotonic sequence or not.
Sol. This is the decreasing sequence, hence we can say that this is monotonic for every n.
As we can notice here that the terms in this sequence will be either zero or negative , so that the sequence is bounded above.
Suppose m = 0 then,
This sequence is not bounded below ,
Hence we can not say that the sequence is bounded.
Question-4: check whether the series 
is convergent or divergent. Find its value in case of convergent.
Sol. As we know that,
Sn = 
Therefore,
Sn = 
Now find out the limit of the sequence,
= ∞
Here the value of the limit is infinity, so that the series is divergent as sequence diverges.
Question-5: : check whether the series 
is convergent or divergent.
Sol. The general formula can be written as,
We get on applying limits,
This is the convergent series and its value is 3 / 4
Question-6: Show that the harmonic series 
is divergent.
Sol. First we will expand its three terms ,
= 1+ 
+ 
We get,
Here if we subtract a finite number from a divergent series , then the resulting series will not be affected by this number.
Therefore we can say that 
is the divergent series which is also a harmonic series.
Question-7: determine the following series is convergent or divergent.
Sol. By using integral test
= ∞
By the integral test, given series is divergent as 
Question-8: Test the series by integral test- 
Sol. Here 
is positive and decreases when we increase n ,
Now apply integral test,
Let,
X = 1 , t = 5 and x = ∞ , t = ∞,
Now,
So by integral test,
The series is divergent.
Question-9: Test the convergence of the following series:
Sol.
Here we take,
Which not zero and finite ,
So by comparison test , 
and 
both converges or diverges, but by p-series test
Is convergent . So that 
is convergent.
Question-10 Test the convergence of the series:
Sol.
By p-series test,
Is convergent ( p = 3>1).
Question-11: : Test the series:
Sol. The series is,
Now,
Take,
Which is finite and not zero.
Which is finite and not zero.
By comparison test 
and 
converge or diverge together.
But,
Is divergent. ( p = ½)
So that 
is divergent.
Question-12: Test the series for convergence.
Sol. For all n>= 0 let,
And let,
For all n>0, since
It follows,
By limit comparison test , the series is divergent
Question-13: : Test the convergence of the following series.
Sol. Neglecting the first term the series can be written as,
So that,
By ratio test 
converges if |x|<1 and diverges if |x|>1, but if |x| = 1 the the test fails,
Then ,
By Raabes’s test 
converges hence the given series is convergent when |x|≤ 1 and divergent If |x| >1.
Question-14: Test the nature of the following series:
Sol.
By ration test 
is convergent when (x/4)<1 and divergent when x>4, when x= 4,
The rario test fails, then
By Rabee’s test 
is convergent , hence the given series is convergent when x<4 and divergent If x >=4.
Question-15: Test the convergence of the series whose nth term is given below-
Sol.
By root test 
is convergent.
Question-16: Test for absolute convergence:
Sol. Let the series is 
,
By ratio test,
is convergent , if |x|<1.
is absolutely convergent if |x|< 1.
Question-17: Prove that 
Solution:
Here f(x) = x cosec x
= 
Now we know that
Question-18: Show that 
Solution:
Here 
Thus
Question-19: Expand 
in power of (x – 3)
Solution:
Let 
Here a = 3
Now by Taylor’s series expansion,
… (1)
equation (1) becomes.
Question-20: Using Taylors series method expand
in powers of (x + 2)
Solution:
Here 
a = -2
By Taylors series,
… (1)
Since
,
, …..
Thus equation (1) becomes
