Questions on sequence and series for practice

Assignment- 07 (B.Tech Sem-I) Mathematics I
Department of Mathematics,
1.Define the terms with one example of each: (a) Bounded sequence, (b) Convergent
sequence, (c) Subsequence, (d) Cauchy sequence, (e) Divergent sequence, (f) Cauchy’s
First theorem on limits, (g) Cauchy convergence criterion, (h) Alternating series, (i)
Alternating series test(Leibnitz test).
2. Check the convergence of the following sequences. Find also its limit, if it exists
𝟏 𝟏 𝟐𝐒𝐧 +𝟏
(a)√𝐧 + 𝟏 − √𝐧 , n∈ ℕ; (b) 𝐧𝐩 , 𝐩 > 0; (c) 𝐒𝟏 = 𝟐, 𝐒𝐧+𝟏 = , n∈ ℕ; (d) 𝐒𝟏 = √𝟐,
𝟑
𝟏⁄
𝟏 𝟏⁄ 𝟏⁄ 𝟏⁄ 𝟏⁄ (𝟑𝐧)! 𝐧
𝐒𝐧+𝟏 = √𝟐𝐒𝐧 , n∈ ℕ; (e) (𝟏 + 𝟐 𝟐 +𝟑 𝟑 +𝟒 𝟒 + ⋯+ 𝐧 𝐧) ; (f){(𝐧!)𝟑 } n∈ ℕ;
𝐧
𝟏⁄
𝒏
𝟑𝐒𝐧 +𝟒 𝟐 𝟑 𝟐 𝟒 𝟑 𝐧+𝟏 𝐧 𝟏 𝟏 𝟏
(g) 𝐒𝟏 = 𝟏, 𝐒𝐧+𝟏 = 𝟐𝐒 ; (h) [(𝟏) (𝟐) (𝟑) … . ( ) ] ;(i)1+𝟐+𝟑+.....+𝐧
𝐧 +𝟑 𝐧

Ans: (a) 0, (b) 0, (c) 1, (d) 2, (e) 0, (f) 27, (g)√2, (h) e, (i) Does not converge.
3. Check the convergence of the following series using general principle of convergence:
1 n+1 1
(a) ∑ 4n ; (b) log( ), (c) ∑ n; Ans: : (a) Convergent, (b) Divergent, (c) Divergent
n
1
4. Prove that ∑ np is convergent if p>1 and divergent if p≤1.

5.Test the convergence of the following series:
𝟑 𝟒 𝟓 𝐧𝐩
(I)2+𝟐 𝐱 + 𝟑 𝐱 𝟐 + 𝟒 𝐱 𝟑 + ⋯ … … , 𝐱 > 0; (II) ∑{(𝐧𝟑 + 𝟏)𝟑 − 𝟏}; (III) ∑ (𝟏+𝐧)𝐪 ;
−𝟏 −𝟐 −𝟑
𝟐𝟐 𝟐 𝟑𝟑 𝟑 𝟒𝟒 𝟒 𝟑 𝟖 𝟏𝟓
(IV) (𝟏𝟐 − 𝟏) + (𝟐𝟑 − 𝟐) + (𝟑𝟒 − 𝟑) + ⋯. ; (V) 𝐱 + 𝟓 𝐱 𝟐 + 𝟏𝟎 𝐱 𝟑 + 𝟏𝟕 𝐱 𝟒 + ⋯ ..

𝟏 −𝐧𝟑/𝟐 𝟏
(VI) ∑ (𝟏 + ) ;(VII) ∑ (𝐧 𝐥𝐨𝐠 𝐧)𝐩 , 𝐧 ≥ 𝟐 (VIII) ∑ 𝐱 𝐧 𝐥𝐨𝐠 (𝐧𝐱);
√ 𝐧

𝟏 𝐏 𝟏.𝟑 𝐏 𝟏.𝟑.𝟓 𝐏 𝟏𝟐 𝟏𝟐 .𝟑𝟐 𝟏𝟐 .𝟑𝟐 𝟓𝟐
(IX) 𝟏𝐏 + (𝟐) +(𝟐.𝟒) + (𝟐.𝟒.𝟔) + ⋯ …; (X) 𝟐𝟐 + 𝟐𝟐 .𝟒𝟐+𝟐𝟐 .𝟒𝟐 𝟔𝟐+......

Ans:(I) Con. If x<1 and div. If x≥1. (II) Con.(III) con. If p-q+1<0 and div. If p-q+1≥0;
(IV) Con. (V) Con If x<1 and div If x≥1, (VI) Conv, (VII)con if p>1 and div if p≤1,
(VIII) Con. If x<1 and div. If x≥1, (IX) Con if p>2 and div if p≤2, (X) Div
1
6. Apply Cauchy Integral test on the series: (a) ∑ n( log n)p , n ≥ 2 ; Ans: Con if p>1 and
1
div if 0≤p≤1; (b) ∑ , Ans: Conergent
n√n2 −1

7.Test the Convergence\ Abs. Convergence and Cond. Convergence of the following
series:
𝟏𝟐 𝟏𝟐 .𝟑𝟐 𝟏𝟐 .𝟑𝟐 𝟓𝟐 𝐧𝟐 𝐬𝐢𝐧 𝐧𝛂
(a)𝟐𝟐 − 𝟐𝟐 .𝟒𝟐 +𝟐𝟐 .𝟒𝟐 𝟔𝟐 -...... (b) ∑(−𝟏)𝐧−𝟏 (𝐧+𝟏)! ; (c) ∑(−𝟏)𝐧 , 𝛂∈ℝ
𝐧𝟑

Ans: (a) Cond. Conv.; (b) Abs. Con. ; (c) Convergence.