Infinite Series MCQ Quiz - Objective Question with Answer for Infinite Series - Download Free PDF

Calculation:

We have,

1/1⁴ + 1/2⁴ + 1/3⁴ + 1/4⁴ + ⋯ = π⁴ / 90         (1)

This can be written as:

[1/1⁴ + 1/3⁴ + 1/5⁴ + ⋯ ] + 1/2⁴ + 1/4⁴ + ⋯ = π⁴ / 90

[1/1⁴ + 1/3⁴ + 1/5⁴ + ⋯ ] + (1/2⁴) × [1/1⁴ + 1/2⁴ + 1/3⁴ + ⋯ ] = π⁴ / 90

Using (1):

[1/1⁴ + 1/3⁴ + 1/5⁴ + ⋯ ] + (1/2⁴) × (π⁴ / 90) = π⁴ / 90

⇒ 1/1⁴ + 1/3⁴ + 1/5⁴ + ⋯ = (π⁴ / 90) × [1 − 1/2⁴] = (π⁴ / 90) × (15/16) = π⁴ / 96

Concept:

Apply the formula 

Calculation:

We have 

Now put x = 1 then 

Hence the option (3) is correct.

Concept:

Expansion of exponential function.

From the above formula value of e¹ and e^-1 are. 

Calculation:

Given expansion is:

we can verify the options and conclude this

Option (A) 5e - 1

not equal to the given series.

option (B) 5e - 3

not equal to given series.

Option (c) 4e

not equal to given series

The answer is option D.

None of the above expansions are matching to given series.

multiply bu -x² on both sides

Adding (1) and (2)

From (3) and (4)

Concept:

Expansion of exponential function.

From the above formula value of e¹ and e^-1 are. 

Calculation:

Given expansion is:

we can verify the options and conclude this

Option (A) 5e - 1

not equal to the given series.

option (B) 5e - 3

not equal to given series.

Option (c) 4e

not equal to given series

The answer is option D.

None of the above expansions are matching to given series.

multiply bu -x² on both sides

Adding (1) and (2)

From (3) and (4)

Concept:

Expansion of exponential function.

From the above formula value of e¹ and e^-1 are. 

Calculation:

Given expansion is:

we can verify the options and conclude this

Option (A) 5e - 1

not equal to the given series.

option (B) 5e - 3

not equal to given series.

Option (c) 4e

not equal to given series

The answer is option D.

None of the above expansions are matching to given series.

Concept:

Apply the formula 

Calculation:

We have 

Now put x = 1 then 

Hence the option (3) is correct.

multiply bu -x² on both sides

Adding (1) and (2)

From (3) and (4)

Calculation:

We have,

1/1⁴ + 1/2⁴ + 1/3⁴ + 1/4⁴ + ⋯ = π⁴ / 90         (1)

This can be written as:

[1/1⁴ + 1/3⁴ + 1/5⁴ + ⋯ ] + 1/2⁴ + 1/4⁴ + ⋯ = π⁴ / 90

[1/1⁴ + 1/3⁴ + 1/5⁴ + ⋯ ] + (1/2⁴) × [1/1⁴ + 1/2⁴ + 1/3⁴ + ⋯ ] = π⁴ / 90

Using (1):

[1/1⁴ + 1/3⁴ + 1/5⁴ + ⋯ ] + (1/2⁴) × (π⁴ / 90) = π⁴ / 90

⇒ 1/1⁴ + 1/3⁴ + 1/5⁴ + ⋯ = (π⁴ / 90) × [1 − 1/2⁴] = (π⁴ / 90) × (15/16) = π⁴ / 96