List of top Mathematics Questions on Convergence tests

Which of the following statements are correct:
A. If $\sum a_n$ converges, then $\lim_{n \to \infty} a_n = 0$ B. If $|a_n| \le c_n$ for all $n$, and if $\sum c_n$ diverges, then $\sum a_n$ diverges.
C. If $|a_n| \le c_n$ for $n \ge N_0$, ($N_0$ is fixed integer) and if $\sum c_n$ converges, then $\sum a_n$ converges.
D. If $a_n \ge d_n \ge 0$ for $n$, and if $\sum d_n$ converges, then $\sum a_n$ converges.
E. If $a_n \ge d_n \ge 0$ for $n \ge N_0$ and if $\sum d_n$ diverges, then $\sum a_n$ diverges.
Choose the correct answer from the options given below:

If \(f(x)=|\sin x|\) for \(-\pi<x<\pi\) and the Fourier series of \(f(x)\) is \[ f(x)=\sum_{n=0}^{\infty}(a_n\cos nx+b_n\sin nx), \] then the value of the Fourier coefficient \(a_0\) is

If \[ u_n=\frac{\sqrt{2n^2-5n+1}}{4n^3-7n^2+2} \] and \[ v_n=\sqrt{n^2+1}-\sqrt{n^2-1}, \] then

A convergent series from the given series is

If \[ u_n=\frac{\sqrt{2n^2-5n+1}}{4n^3-7n^2+2} \] and \[ v_n=\sqrt{n^2+1}-\sqrt{n^2-1}, \] then

If \(f(x)=|\sin x|\) for \(-\pi<x<\pi\) and the Fourier series of \(f(x)\) is \[ f(x)=\sum_{n=0}^{\infty}(a_n\cos nx+b_n\sin nx), \] then the value of the Fourier coefficient \(a_0\) is

A convergent series from the given series is

If \(R\) denotes the radius of convergence of the power series

Which one of the following series is convergent?

Let \( \sum_{n \geq 1} a_n \) be a convergent series of positive real numbers. Then which of the following statement(s) is (are) true?

Let \( \{a_n\}_{n \geq 1} \) be a sequence of real numbers such that \[ a_n = \frac{1 + 3 + 5 + \dots + (2n - 1)}{n!}, n \geq 1. \] Then \( \sum_{n \geq 1} a_n \) converges to ................

Which of the following is TRUE?

The sum of the series \[ \sum_{n=1}^{\infty} \tan^{-1} \left( \frac{2}{n^2} \right) \] is

The interval of convergence of the power series \[ \sum_{n=1}^{\infty} \frac{1}{(-3)^n + 2} \frac{(4x - 12)^n}{n^2 + 1} \] is