Question:
Evaluate the infinite series:
\[
1 + \frac{1}{3} + \frac{1.3}{3.6} + \frac{1.3.5}{3.6.9} + \dots \text{ to } \infty =
\]
Evaluate the infinite series:
\[
1 + \frac{1}{3} + \frac{1.3}{3.6} + \frac{1.3.5}{3.6.9} + \dots \text{ to } \infty =
\]
Show Hint
For infinite series involving factorial and double factorial patterns, recognize standard summation results to quickly evaluate their limits.
Updated On: Mar 25, 2025
- \( \sqrt{5} \)
- \( \sqrt{6} \)
- \( \sqrt{15} \)
- \( \sqrt{3} \)
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
The given series is:
\[
S = 1 + \frac{1}{3} + \frac{1.3}{3.6} + \frac{1.3.5}{3.6.9} + \dots \text{ to } \infty.
\]
Step 1: Identifying the pattern
Observing the general term:
\[
T_n = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{3 \cdot 6 \cdot 9 \cdots (3n)}.
\]
This is a standard series expansion for the function:
\[
\sum_{n=0}^{\infty} \frac{(2n-1)!!}{(3n)!!}.
\]
From known mathematical results, the sum of the given infinite series converges to:
\[
\sqrt{3}.
\]
Step 2: Conclusion
Thus, the given series evaluates to:
\[
\boxed{\sqrt{3}}.
\]
Was this answer helpful?
0
0
Top Questions on Series
- The value of \[ \lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k + 3)!} \] is:
- In the expansion of $(1 + x)^n$, $\frac{{}^nC_1}{{}^nC_0} + 2 \cdot \frac{{}^nC_2}{{}^nC_1} + 3 \cdot \frac{{}^nC_3}{{}^nC_2} + \dots + n \cdot \frac{{}^nC_n}{{}^nC_{n-1}}$ is equal to:
- If \( \sum_{k=1}^{n} k(k+1)(k-1) = pn^4 + qn^3 + tn^2 + sn \), where \( p, q, t, s \) are constants, then the value of \( s \) is equal to:
- If \( A = 1 + r^a + r^{2a} + r^{3a} + \dots \infty \) and \( B = 1 + r^b + r^{2b} + r^{3b} + \dots \infty \), then \( \frac{a}{b} \) is equal.
- The sum of the infinite series \(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \dots\) is equal to:
View More Questions
Questions Asked in AP EAPCET exam
- In Young’s double slit experiment, two slits are placed 2 mm from each other. The interference pattern is observed on a screen placed 2 m from the plane of the slits. Then the fringe width for a light of wavelength 400 nm is:
- AP EAPCET - 2024
- Wave optics
- Any magnetic material loses its magnetic property when it is:
- AP EAPCET - 2024
- Magnetic Field
- The average value of electric energy density in an electromagnetic wave is: [where \( E_0 \) is the peak value]
- AP EAPCET - 2024
- Electromagnetic waves
- A resistance of \( 20 \Omega \) is connected to a source of an alternating potential \( V = 200 \sin(10\pi t) \). If \( t \) is the time taken by the current to change from the peak value to the rms value, then \( t \) is (in seconds):
- AP EAPCET - 2024
- Alternating current
- When two coaxial coils having the same current in the same direction are brought close to each other, then the value of current in both the coils:
- AP EAPCET - 2024
- Electromagnetic induction
View More Questions