The sequence \(\big\{ a_{n= \frac{1}{ n^{2} }; n>0 } \big\} \) is
Sequence Behavior for \( a_n = \frac{1}{n^2} \)
We are asked to determine the behavior of the sequence defined by \( a_n = \frac{1}{n^2} \) for \( n > 0 \).
The terms of the sequence are generated by substituting positive integer values for \( n \). Let's look at the first few terms:
- For \( n=1 \), \( a_1 = \frac{1}{1^2} = 1 \)
- For \( n=2 \), \( a_2 = \frac{1}{2^2} = \frac{1}{4} \)
- For \( n=3 \), \( a_3 = \frac{1}{3^2} = \frac{1}{9} \)
- For \( n=4 \), \( a_4 = \frac{1}{4^2} = \frac{1}{16} \)
- ...and so on.
To understand the behavior of the sequence as \( n \) gets larger, we need to examine its limit as \( n \) approaches infinity (\( n \to \infty \)).
Calculating the Limit of the Sequence
A sequence is defined as convergent if its terms approach a specific finite value as \( n \) tends towards infinity. This value is called the limit of the sequence. We calculate this limit for the given sequence:
$ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n^2} $
As \( n \) becomes larger and larger (e.g., 10, 100, 1000, ...), \( n^2 \) grows even faster (100, 10000, 1000000, ...). Consequently, the fraction \( \frac{1}{n^2} \) becomes smaller and smaller, getting arbitrarily close to 0.
Therefore, the limit is:
$ \lim_{n \to \infty} \frac{1}{n^2} = 0 $
Determining Sequence Convergence
Since the limit of the sequence \( a_n = \frac{1}{n^2} \) exists and is equal to a finite number (0), the sequence is classified as convergent.
The sequence is not:
- Divergent: A sequence diverges if its limit is infinite (\( \infty \) or \( -\infty \)) or does not exist.
- Oscillating: A sequence oscillates if its terms do not settle down towards a single value but fluctuate indefinitely.
Because \( \lim_{n \to \infty} \frac{1}{n^2} = 0 \), the sequence clearly converges to 0.
Important Questions from Algebra (Notes)
- What is the remainder when 2023²⁰²⁴ + 2025²⁰²⁴ is divided by 2024?
In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. If she/he attempts all 60 questions and secures 130 marks, the number of questions she/he attempts wrongly, are?
Match List-I with List-II
List-1 List-II (A) If $\begin{bmatrix}\lambda-1 & 0 \\ 0 & \lambda-1 \end{bmatrix} $, then $\lambda$ is (I) 0 (B) If A=$ \begin{bmatrix}1 & 2 \\2 & 4 \end{bmatrix} $, then $\Delta$ is (II) 1 (C) If A = $ \begin{bmatrix}1 & 0 \\0 & \frac{1}{2} \end{bmatrix} $, then $|A^{-1}|$ is (III) -2 (D) If $ \begin{bmatrix}a+1 & 1 \\1 & 2 \end{bmatrix} = \begin{bmatrix}-1 & 1 \\1 & 2 \end{bmatrix} $, then a is (IV) 2 Choose the correct answer from the options given below:
- If (x - 1) is a factor of $2x^2 - 5x + k = 0$, then the value of k is:
- If $x = (2+\sqrt{3})^{\frac{1}{3}} + (2+\sqrt{3})^{-\frac{1}{3}}$ and $x^3-3x + k = 0$, then the value of k is:
