∑n = 1∞2n2 + n + 1n! is equal to
2e - 1
2e + 1
6e - 1
6e + 1
If a < 1, b = ∑k = 1∞akk, then a is equal to
∑k = 1∞- 1kbkk
∑k = 1∞- 1k - 1bkk!
∑k = 1∞- 1kbkk - 1!
∑k = 1∞- 1k - 1bkk + 1!
1 + 24 + 24 58 + 24 58 812 + 24 58 812 1116 + . . . . is equal to:
4 - 23
163
43
432
If x < 1 and y = x - x22 + x33 - x44 + . . . , then x is equal to
y + y22 + y33 + . . .
y - y22 + y33 - y44 + . . .
y + y22! + y33! + . . .
y - y22! + y33! - y44! + . . .
If Sn = 13 + 23 + ... + n3 and Tn = 1 + 2 + ... + n, then
Sn = Tn3
Sn = Tn2
For any integer n ≥ 1, the sum ∑k = 1nkk + 2 is equal to
nn + 1n + 26
nn + 12n + 16
nn + 12n + 76
nn + 12n + 96
If 1 + x + x2 + x35 = ∑k = 015akxk, then ∑a2k = 07k = 0
128
256
512
1024
If α = 52 ! 3 + 5 . 73 ! 32 + 5 . 7. 94! 33 + . . . , thenα2 + 4α is equal to
21
23
25
27
11 . 3 + 12 . 5 + 13 . 7 +14 . 9 + . . . = ?
2loge2 - 2
2 - loge2
2loge4
loge4
B.
Let S = 11 . 3 + 12 . 5 + 13 . 7 +14 . 9 + . . . Tn = 1n2n + 1 = 1n - 22n + 1⇒ S = ∑n = 1∞Tn = ∑n = 1∞1n - 22n + 1 = 11 - 23 + 12 - 25 + 13 - 27 + 14 - 29 + 15 - . . . = 1 + 12 - 13 + 14 - 15 + . . . = 1 - - 12 + 13 - 14 + 15 - . . . = 1 - - 1 + loge2 = 2 - loge2
If l, m, n are in arithmetic progression, then the straight line b + my + n = 0 will pass through the point
(- 1, 2)
(1, - 2)
(1, 2)
(2, 1)