1) Find the sum of the following series to Infinity::
a)1-1/3+1/3²+1/3³+1/3⁴+... 3/4
b) 2/5+3/5²+2/5³+3/5⁴... 13/24
c) 8+4√2+4+...... 8(2+√2)
d) 9¹⁾³.9¹⁾⁹.9¹⁾²⁷....... 3
e) 2¹⁾⁴.4¹⁾⁸.8¹⁾¹⁶.16¹⁾³².... 2
2) If Sₚ denotes the sum of the series 1+ rᵖ+r²ᵖ+...to inf. and sₚ the sum of the series 1- rᵖ+ r²ᵖ - ...to inf., Prove Sₚ + sₚ= 2.S₂ₚ
3) find the sum of the terms of an infinite decreasing GP in which all the terms are positive, the first term is 4, and the difference between third and fifth term is equals to 32/81. 6, 12/(3-2√2)
4) find an Infinity GP whose first term is 1 and each term is the sum of all the terms which follow it. 1,1/2,1/4.......
5) the sum of the first two terms of an infinite GP is 5 and each term is three times the sum of the succeeding terms. find the GP.
4,1,1/4, 1/16,......
6) Express the recurring decimal 0.125125125....as a rational number. 125/999
7) find the rational number whose decimal expansion is 0.423232323... 419/990
8) show that in an infinite GP with common ratio r(|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
9) one side of an equilateral triangle is 18cm. the midpoints of its sides are joined to form another triangle whose midpoints, in terms are joined to form another triangle, the process is continued in definitely. find the sum of the
a) perimeter of all the triangles
b) areas of all triangles. 108,108√3
10) If S denotes the sum of an infinite GP M denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively 2SM/(S²+M) and (S²-M)/(S²+M)
11) the inventor of the chess board suggested a reward of grain of wheat for the first square, 2 grains for the second, 4 grains for the third and so on, doubling the number of the grains for subsequent squares. How many grains would have to be given to inventor? (There are 64 squares in chess board). 2⁶⁴-1
** Find the sum of the following series to n terms
12) 2²+ 4²+6²+8²+...... 2n/3 (n+1)(2n+1)
13) 1³+3³+5³+7³+... n²(2n²-1)
14) 2³+4³+6³+8³+... 2(n(n+1))
15) 1.2²+2 3²+3.4²+... n/2(n+1)(n+2)(3n+5)
16) 1.2.5+2.3.6+3.4.7+... n/12 (n+1)(3n²+23n+34)
17) 1.2.4+2.3.7+3.4.10+... n/12 (n+1)(9n²+25n+14)
18) 1+(1+2)+(1+2+3)+(1+2+3+4) +... n/9 (n+1)(n+2)
19) 1x2 + 2x 3 + 3x4 + 4x 5+.... n/3(n+1)(n+2)
20) 3x 1²+ 5x2²+ 7x3²+ ..... n/6 (n+1)(3n²+5n+1).
21) 2n²- 3n+5. n/6 (4n²-3n+23)
22) 2n³+ 3n² - 1. n/2 (n³+ 4n² +4n-1)
23) n³ - 3ⁿ. {n(n+1)/2}² -3/2 (3ⁿ-1)
24) n(n+1)(n+4). n(n+1)/12 (3n²+23n+34)
25) (2n-1)². n/3 (2n²+1)(2n-1)
26) 3+5+9+15+23+... n/3 (n²+8)
27) 2+5+10+17+26+... n/6 (2n²+3n+7)
28) 1+3+7+13+21+... n/3 (n²+2)
29) 3+7+14+24+37+... n/2 (n²+n+4)
30) 1+3+6+10+15+... n/6 (n+2)(n+2)
31) 1+4+13+40+121+... n/4 (3ⁿ⁺¹- 2n -3)
32) 3+18+57+132+255+ ... n/2 (n+1) (n²+n+1)
33) 4+6+9+13+18+.. n/6 (n²+3n+20)
34) 2+10+30+68+130+... n/4 (n+1)(n²+n+2)
35) 2+4+7+11+16+... n/6 (n²+3n+8)
36) 1/1.4+ 1/4.7 +1/7.10+... n/(3n+1)
37) 1/1.6 + 1/6.11 +1/11.14+ 1/14.19+... 1/{(5n-4)(5n+1)}. n/(5n+1)
38) If the sum of first n even natural number is equal to k times the sum of first n odd natural numbers, then k =
A) 1/n B) (n-1)/n C) (n+1)/2n D) (n+1)/n.
38) The sum of the series 1/log₂4 + 1/log₄4 + 1/log₈4 +.....1/log₂ⁿ 4 is..
A) n(n+1)/2 B) n(n+1)(2n+1)/2
C) n(n+1)/4. D) none
39) the value of ⁿᵣ₌₁ ∑(2r-1)a + 1/bʳ) is equal to
A) an²+ (bⁿ⁻¹ - 1)/(bⁿ⁻¹(b-1)
B) an²+ (bⁿ - 1)/{bⁿ(b-1)}.
C) an³ + (bⁿ⁻¹ -1)/{bⁿ(b-1)} D) none
40) If ∑n= 210, ∑n²=
A) 2870. B) 2160 C) 2970 D) n
41) If Sₙ= ⁿᵣ₌₁ ∑{1+2+2²+... Sum to r terms)/2ʳ then Sₙ is equals to
A) 2ⁿ - n - 1 B) 1 - 1/2ⁿ
C) n-1 + 1/2ⁿ. D) 2ⁿ - 1
42) If 1 + (1+2)/2 + (1+2+3)/3 + ....n terms is S, then S is equal to
A) n(n+3)/4. B) n(n+2)/4
C) n(n+1)(n+2)/6 D) n²
43) sum of n terms of the series √2+ √8+ √18+ √32+... is
A) n(n+1)/2 B) 2n(n+1)
C) n(n+1)/√2. D) 1
44) The sum of 10 terms of the series √2+ √6+ √18+... is
A) 121(√6+√2). B) 243(√3+1)
C) 121/(√3 -1) D) 242(√3-1)
45) the sum of the series 1²+3²+5²+... to n terms is
A) n(n+1)(2n+1)/2
B) n(2n-1)(2n+1)/3.
C) (n-1)²(2n+1)/6
D) (2n+1)³/3
46) The sum of the series 1+ 2.2 + 3.2²+ 4.2³+ 5 2⁴+ 100.2⁹⁹ is
A) 99x 2¹⁰⁰ B) 99x 2¹⁰⁰ +1.
C) 100 x 2¹⁰⁰ D) none
47) the sum to n terms of the series 1/(√1+√3) + 1/(√3+ √5) + 1/(√5+ √7) + ....+... is
A) √(2n+1) B) 1/2 √(2n+1)
C) √(2n+1) - 1 D) 1/2 √(2n+1) -1.
48) The sum of the series 2/3 + 8/9 + 26/27+ 80/81+....to n terms is
A) n - 1/2(3⁻ⁿ -1)
B) n - 1/2(1 -3⁻ⁿ).
C) n + 1/2(3ⁿ -1)
D) n - 1/2(3ⁿ -1)
49) Define the number e. Show that its value lies between 2 and 3.
50) find the coefficient of x¹⁰ in the expansion of e²ˣ. 2¹⁰/10!
51) find the coefficient of x² in the expansion of x³ˣ⁺⁴. 9e⁴/2
52) find the coefficient of x⁵ in the expansion of (1- 2x +3x²)/eˣ. -71/120
53) find the coefficient of xⁿ in the expansion of (a+bx+cx²)/eˣ. (-1)ⁿ/n! . [cn² - (b+c)ⁿ+a]
54) Expand:
a) (e⁴ˣ - 1)/e²ˣ in the ascending power of x. 2[2x+(2x)³/3! + (2x)⁵/5! +.......]
b) Expand (e⁷ˣ + eˣ)/2e⁴ˣ in the ascending power of x. (1+ 3²x²)2! + 3⁴x⁴/4! + 3⁶x⁶/6!+.....)
c) 1)2[ₑx√-1 + ₑ-x√-1].
55) Show that the coefficient of xⁿ in the series. [(1+x)/1! + (1+x)²/2! + (1+x)³/3!+........+........] is e/n!
56) Show that (eⁱˣ + e⁻ⁱˣ)/2 = (1 - x²/2!+ x⁴/4! - ........)
** PROVE::
56)
a) {(log 2) + (log 2)²/2! + .....}= 1
b) {1+ a²x²/2! + a⁴x⁴/4! +....}² -{ax +a³x³/3! + a⁵x⁵/5! + .....}²= 1
c) {(e²+1)/(e² -1)} = {1+ 1/2! +1/4! +1/6! +....}/{1+ 1/3!+ 1/5!+ 1/7! +....}.
d) (1+ 1/2! +1/4! +1/6!+....}² - {1+ 1/3! + 1/5! +....}² = 1
e) 1+[{1+ 2²/2! +2⁴/3!+ 2⁶/4!+....}/ {1+ 1/2! +2/3! + 2²/4!+....}] = e².
f) {2/1! + 4/3! + 6/5! +....} = e.
g) {1+ 3/1!+ 5/2! +7/3!+....}= 3e
h) {1+ (1+2)/1! + (1+2+3)/2!+....}= 7e/2
I) {1/1! + (1+3)/2! + (1+3+5)/3! + ....} = 2e
j) {2/1! + (2+4)/2!+(2+4+6)/3!+ ....} = 3e.
k) {3+ 5/1!+7/2!+9/3!+....}= 5e.
l) {1²+ 2²/2!+3²/3!+4²/4!+....}= 2e.
m) {2/1! + 4/3! + 6/5! +.....}= {1/2/3! +4/5! + 6/7! + ......}
57) Find the sum of the following series:::
a) {1+ (1+2)/2! + (1+2+2²)/3!+ (1+2+2²+2³)/4! +....} e² - e
b) {1+ (1²+2²)/2! + (1²+2²+3²)/3!+ (1²+2²+3²+4²)/4! +....} 17e/6
c) {1+ 1/2! + 1.3/4!+ 1.3.5/6! +....}. √e
d) {1/1.2 + 1.3/1.2.3.4 + 1.3.5/1.2.3.4.5.6 +....} √e - 1
58) Find The Value of:::
a) √e up to four places of decimals. 1.6486
b) 1/√e up to four places of decimal. 0.6065
c) e⁻¹⁾⁸ up to three places of decimal. 0.819
59) If | x | < 1, prove that:
a) 1/2 log{(1+x)/(1-x)}= (x + x³/3 + x⁵/5 + .........)
b) {x²/2 + 2x³/3 + 3x⁴/4 + 4x⁵/5 + ...} = {x/(1-x) + log(1-x)}
60) Prove:
a) log m - log n= 2.[{(m-n)/(m+n)} + 1/3. {(m-n)/(m+n)}³ + 1/5. {(m-n)/(m+n)}⁵ +..........]
b) log {(m+n)/(m-n)}= [2mn/(m²+n²) + 1/3. {(2mn)/(m²+n²)}³ +.........]
c) log m= 2[{(m-1)/(m+1)} + 1/3. {(m-n)/(m+n)}³+ 1/5. {(m-n)/(m+n)}⁵ +.......]
d) log(1+x+x²+......)= (x + x²/2 + x³/3 + x⁴/4 +........)
e) 2 log x - log(x+1) - log(x-1) = [1/x² + 1/2x⁴ + 1/3x⁴ +......]
f) [1/2 - 1/2.2² + 1/3.2³ + 1/4.2⁴ +...] = Log 3 - log 2
g) [1/1.2 - 1/2.3 + 1/3.4 + 1/4.5 +...] = 2Log 2 - 1
h) log(x+1) - logx= 2.[1/(2x+1) + 1/3+2x+1)³ + 1/5(2x+ 1)⁵ +...] =
I) [1/2 +(1/2 +1/3) - 1/4.(1/2²+ 1/3²) + 1/6. (1)2³ + 1/3³) +......] = Log √2
j) [1/1.2.3 + 5/3.4.5+ 9/5.6.7 +...] = 5/2 - 3log 2 .
61a) If y= (x + x²/2 + x³/4 + .........) Then show x= (1- e⁻ʸ)
b) If m,n are the roots of the Equation x²- px +q= 0 prove that log(1+ px+ qx²)= [(m+n)x - {(m²+n²)/2}x² + {(m³+ n³)/3}x³ - ....}
62) Find the sum of the series and prove that:
a)
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