| Q.29 The value of the series 1 + sin x + cos2 x + sin3 x + β¦ at x = Ο/4 is ______. | |
| (A) | 1ββ2 + 1 |
| (B) | β2ββ2 + 1 |
| (C) | 1ββ2 β 1 |
| (D) | β2ββ2 β 1 |
Question
The value of the series
1 + sin x + cos 2x + sin 3x + β¦
at
x = Ο/4
is ______.
Step-by-Step Solution
Step 1: Substitute the value of x
At x = Ο/4,
sin(Ο/4) = cos(Ο/4) = 1/β2
Step 2: Rewrite the series
Original series:
1 + sin x + cos 2x + sin 3x + β¦
Substituting values:
= 1 + 1/2 + (1/2)2 + (1/2)3 + β¦
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 β r)
Substituting values:
S = 1 / (1 β 1/2)
Multiply numerator and denominator by 2:
S = 2 / (2 β 1)
Final Answer
2 / (2 β 1)
Correct Option
β Option (D)
Explanation of All Options
Option (A)
1/2 + 1/2 + 1
β Incorrect β does not match the infinite GP sum.
Option (B)
2 / (2 + 1)
β Incorrect β denominator sign is incorrect.
Option (C)
1 / (2 β 1)
β Incorrect β numerator is missing.
Option (D)
2 / (2 β 1)
β Correct β exact value obtained using the GP formula.
Conclusion
The given series forms an infinite geometric progression at x = Ο/4 with a common ratio less than 1.
Applying the infinite GP sum formula gives the correct value:
2 / (2 β 1)
This type of question is frequently asked in JEE, NDA, CUET, and other competitive exams, making it essential to quickly recognize series patterns.
