Problem 467 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{n \rightarrow \infty} \frac{1-2+3-4+\cdots \cdots \cdots+(2 n-1)-2 n}{\sqrt{n^{2}+1}} $$

Step-by-Step Solution

Answer

The value of the given limit is 1.

Step 1: Simplify the numerator

The numerator is a sequence of alternating-addition and subtraction. We notice that it alternates between even and odd numbers. In general, we can represent this as \( sum = 1 - 2 + 3 - 4 + \cdots + (2n-1) - 2n \). When we separate the positive odd terms and negative even terms, we get \( sum = (1 + 3 + 5 + \cdots + (2n-1)) - (2 + 4 + 6 + \cdots + 2n) \). Hence, we can simplify the numerator as \( sum = n^2 - n \).

Step 2: Plug the simplified numerator into the function

Now, we substitute the simplified numerator into the function. We get \( \lim _{n \rightarrow \infty} \frac{n^2 - n} {\sqrt{n^2 + 1}} \).

Step 3: Divide the numerator and denominator by \( n^2 \)

To simplify the equation further, we divide the numerator and denominator by \( n^2 \). The function becomes \( \lim _{n \rightarrow \infty} \frac{1 - \frac{1}{n}}{\sqrt{1 + \frac{1}{n^2}}}\).

Step 4: Apply the limits

Now, as \( n \rightarrow \infty \), \( \frac{1}{n} \) and \( \frac{1}{n^2} \) will tend towards zero because as the denominator increases the value of whole fraction decreases. So, our limit equation simplifies to \( \frac{1 - 0}{ \sqrt{1 + 0}} \).

Key Concepts

Infinite Series

Infinite series can be a challenging yet intriguing concept. They involve a sequence of numbers that continues indefinitely, often governed by a specific rule. These series can often be summed up in closed-form expressions, especially if they follow a recognizable pattern.
In our provided exercise, the sequence goes as follows: 1 - 2 + 3 - 4 + ... + (2n-1) - 2n. This can be grouped into two distinct sequences:

• Positive odd numbers: 1, 3, 5, ..., (2n-1)
• Negative even numbers: -2, -4, -6, ..., -2n

By recognizing the pattern, we simplify the sum of these opposing series using the arithmetic progression formula. This allows for efficient simplification before applying other calculus concepts like limits.

Simplification of Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in calculus and algebra. It involves reducing expressions to their most concise form without changing their value.
In our original exercise, we started with a complex looking numerator: the alternating sum of odd and even numbers. To simplify this, we separated it into two arithmetic sequences as shown:

• The odd sequence sums to a squared term: \( n^2 \)
• The even sequence sums along a linear progression: \( n \)

Hence, \( sum = n^2 - n \). This concise form is crucial, especially when it comes to further operations like limits, as it reduces computational complexity and enhances efficiency.

Applying Limits

Applying limits is a fundamental concept in calculus, often used to determine the behavior of functions as they approach a particular value. Limits are particularly instrumental in handling expressions involving infinity.
By examining our simplified expression \( \frac{n^2 - n}{\sqrt{n^2 + 1}} \), we apply limits by substituting infinity for \( n \). This process reveals how the function behaves into the infinite realm.
To further simplify, consider dividing both the numerator and denominator by \( n^2 \). This results in \( \lim _{n \rightarrow \infty} \frac{1 - \frac{1}{n}}{\sqrt{1 + \frac{1}{n^2}}} \). As \( n \) tends to infinity, small terms like \( \frac{1}{n} \) and \( \frac{1}{n^2} \) both approach zero, further simplifying the task at hand.
This results in the limit simplifying to a clean, final form of \( \frac{1}{1} \), equating to 1, which shows the elegance and power of applied limits in calculus.