Q. 49 - Calculus

Question

\[\sum\limits_{k = 1}^{k = \infty } {\frac{{{k^p}}}{{{e^k}}}} \]

Step-by-Step Solution

Answer

the values of p are 

Step 1. Applying convergent series test

Given series :

\[\sum\limits_{k = 1}^{k = \infty } {\frac{{{k^p}}}{{{e^k}}}} \]

Let us apply ratio convergence  test:

let \[{a_k}\]=\[\sum\limits_{k = 1}^{k = \infty } {\frac{{{k^p}}}{{{e^k}}}} \]

and \[{a_k+1}\] =\[\sum\limits_{k = 1}^{k = \infty } {\frac{{{k+1^p}}}{{{e^k+1}}}} \] 

   

Step 2: Attempt direct substitution

Try substituting the value the variable approaches directly into the expression.

Step 3: Handle indeterminate forms

If direct substitution gives an indeterminate form, apply L'Hopital's Rule, algebraic manipulation, or other techniques.

Step 4: Evaluate the limit

Compute the final value of the limit.

Step 5: State the conclusion

Express the final answer.

Step 6: Conclude with the answer

the values of p are