Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 51

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Let $S$ denotes the infinite sum $2 + 5 x + 9 x^{2} + 14 x^{3} + 20 x^{4} + \dots$ where $| x | < 1$ . Then $S$ equals -

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Important Questions on Sequences and Series

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 52

The coefficient of

x^{101}

in the expansion of

(1 - x) (1 - 2 x) (1 - 2^{2} x) \dots (1 - 2^{101} x)

is -

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 53

The sum of the series

1^{2} + 1 + 2^{2} + 2 + 3^{2} + 3 + \dots + n^{2} + n

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 54

The sum of the series

3 + 7 + 13 + 21 + 31 + \dots

to

n

terms is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 55

The sum of the series

\frac{7}{1^{2} \cdot 6^{2}} + \frac{17}{6^{2} \cdot 11^{2}} + \frac{27}{11^{2} \cdot 16^{2}} + . . .

is

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 56

The sum to 100 terms of the series

1 .2.3. + 2.3.4 . + 3.4.5 . + \dots + n (n + 1) (n + 2) + \dots

is integral multiple of -

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 57

The sum of the first

n

terms of the series

1^{2} + 2 . 2^{2} + 3^{2} + 2 . 4^{2} + \dots . .

is

\frac{n (n + 1)^{2}}{2}

when

n

is even. Then the sum if

n

is odd, is

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 58

If

x, y, z

and

w

are non-zero real numbers and

x^{2} + 5 y^{2} + 5 z^{2} + 4 w^{2} - 4 x y - 4 y z - 4 z w = 0

, then

x, y, z, w

are in

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 59

\sum_{n = 0}^{\infty} \frac{(\ln x)^{n}}{n!}

is equal to