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Q. Evaluate
- Analyze Behavior as Approaches Infinity: We are asked to evaluate the infinite series . This series does not have a straightforward formula for direct evaluation, so we need to analyze its behavior as approaches infinity.
- Consider : Let's consider the term as approaches infinity. We know that as becomes very large, approaches . Therefore, the base approaches .
- Examine Limit Resembling : Now, we need to consider the exponent . As approaches infinity, also approaches infinity. So we have an expression of the form raised to the power of .
- Rewrite Term for Clarity: This expression resembles the limit that defines the number , where is the base of the natural logarithm. The limit is given by . However, in our case, we have a subtraction inside the parentheses, and the exponent is instead of .
- Calculate Limit of : We can rewrite the term as to see the resemblance to the limit more clearly. However, since we have a subtraction, the limit of as approaches infinity is not , but . This is because is the reciprocal of .
- Evaluate as Approaches Infinity: Now, raising to the power of , we get . As approaches infinity, approaches because is less than and any number less than raised to an infinite power tends to .
- Determine Convergence to : Since each term of the series approaches as approaches infinity, the sum of the series is the sum of an infinite number of terms that each approach . This suggests that the series converges to .
- Final Result: Series Sum is : Therefore, the value of the infinite series is .
