Adani University

Mathematics-I
Tutorial 2: Sequence & Series

Sr. Questions
No.
1 Test the convergence of the series
1 1
(1) ∑∞𝑛=3 (2)∑∞
𝑛=1 .
2
(𝑛𝑙𝑜𝑔𝑛)√𝑙𝑜𝑔 𝑛−1 1+22 +32 +⋯+𝑛2

2 Let 𝑆 = ∑∞𝑛=1 𝑛𝛼
𝑛
where |𝛼| < 1. Find the value of 𝛼 in (0,1) such
that 𝑆 = 2𝛼.
3 Check the convergence of the following series
∞ 5𝑛3 −3𝑛 ∞ 2𝑛
(1) ∑𝑛=1 2 (2) ∑𝑛=1 3 .
𝑛 (𝑛−2)(𝑛2+5) 𝑛 +1
1
4 Show that the 𝑝 − 𝑠𝑒𝑟𝑖𝑒𝑠 ∑∞
𝑛=1 𝑛𝑝 (p
is a real constant) converges if
𝑝 > 1 and diverges if 𝑝 ≤ 1.
5 Define the geometric series and find the sum of the following series
∞ 3𝑛−1 −1
∑𝑛=1 𝑛−1
6
6 Test the convergence of the series
𝑥3 𝑥5
𝑥− + − ⋯ , 𝑥 > 0.
3 5
7 Test the convergence of the series ∑∞ 4 4
𝑛=1(√𝑛 + 1 − √𝑛 − 1).
3
8 Test the convergence of the series ∑∞ 3
𝑛=1[ √𝑛 + 1 − 𝑛].
9 10 20 40
Test the convergence of the series 5 − + − +⋯
3 9 27
10 Check the convergence of the following series
2
1 1 𝑛
(1) ∑∞
𝑛=1 (2) ∑∞
𝑛=1 (1 + )
12 +22+⋯+𝑛2 𝑛
11 1 1 1 1
Check the convergence of the following series − + − +⋯
1∙2 3∙4 5∙6 7∙8
12 (𝑙𝑛𝑛)3
Check the convergence of the series ∑∞
𝑛=1 { } and
𝑛3
∑∞ 𝑛
𝑛=1(−1) (√𝑛 + √𝑛 − √𝑛).
13 n2
Investigate the convergence of ∑∞
𝑛=1 .
7n
14 Check the convergence of ∑∞ 𝑛=1(√𝑛 + 1 − √𝑛)
15 Check the absolute and conditional convergence of the series
𝑛2
∑∞
𝑛=1(−1)
𝑛
.
𝑛3 +1
16 Check the absolute and conditional convergence of the series
(−1)𝑛
∑∞
𝑛=1 .
√𝑛+√𝑛+1
17 𝑥 2𝑛−1
For the series ∑∞
𝑛=0(−1)
𝑛−1
, find the radius and interval of
2𝑛−1
convergence.
18 Find the radius of convergence and interval of convergence of the series
(−1)𝑛 (𝑥+2)𝑛
∑∞𝑛=1 . For what values of 𝑥 does the series converge
𝑛
absolutely, conditionally?
n 2
19 Investigate the convergence of ∑∞
𝑛=1 7n .
20 2n (𝑛!)2
Investigate the convergence of ∑∞
𝑛=1 (2n)!
21 Find the radius of convergence and interval of convergence of the series
1 1 1 𝑛
1 − (𝑥 − 2) + (𝑥 − 2 )2 + ⋯ + (− ) (𝑥 − 2)𝑛 + ⋯
2 22 2
22 (3𝑥−2)𝑛
Find radius and interval of convergence for ∑∞𝑛=0 { }. For what
𝑛
values of 𝑥 does the series converge absolutely?
23 ∞ 2n +1
Use the ratio test to check the convergence of the series ∑𝑛=1 n
3 +1
24 2n +5
Check the convergence of ∑∞
𝑛=1 3n
25 Test the convergence of the series
1 𝑥 𝑥2 𝑥3
+ + + + ⋯;𝑥 ≥ 0 .
1∙2∙3 4∙5∙6 7∙8∙9 10∙11∙12
26 1 4
Find the sum of the series ∑𝑛≥2 𝑛
and ∑𝑛≥1
4 (4𝑛−3)(4𝑛+1)