Question

The value of the series

1 + sin x + cos 2x + sin 3x + …

at

x = π/4

is ______.

Step-by-Step Solution

Step 1: Substitute the value of x

At x = π/4,

sin(π/4) = cos(π/4) = 1/√2

Step 2: Rewrite the series

Original series:

1 + sin x + cos 2x + sin 3x + …

Substituting values:

= 1 + 1/2 + (1/2)² + (1/2)³ + …

Step 3: Identify the series pattern

This is an infinite geometric progression (GP) with:

• First term, a = 1
• Common ratio, r = 1/2

Step 4: Apply the infinite GP formula

The sum of an infinite GP is given by:

S = a / (1 − r)

Substituting values:

S = 1 / (1 − 1/2)

Multiply numerator and denominator by 2:

S = 2 / (2 − 1)

Final Answer

2 / (2 − 1)

Correct Option

✅ Option (D)

Explanation of All Options

Option (A)

1/2 + 1/2 + 1

❌ Incorrect — does not match the infinite GP sum.

Option (B)

2 / (2 + 1)

❌ Incorrect — denominator sign is incorrect.

Option (C)

1 / (2 − 1)

❌ Incorrect — numerator is missing.

Option (D)

2 / (2 − 1)

✅ Correct — exact value obtained using the GP formula.

Conclusion

The given series forms an infinite geometric progression at x = π/4 with a common ratio less than 1.
Applying the infinite GP sum formula gives the correct value:

2 / (2 − 1)

This type of question is frequently asked in JEE, NDA, CUET, and other competitive exams, making it essential to quickly recognize series patterns.