To solve the sum [tex]\(\sum_{k=0}^4(1+k!)\)[/tex], we need to calculate each term in the series for [tex]\(k\)[/tex] ranging from 0 to 4, and then add those terms together. Let’s break it down step by step:
1. Understanding the Term:
- Each term in the series is given by [tex]\(1 + k!\)[/tex], where [tex]\(k!\)[/tex] denotes the factorial of [tex]\(k\)[/tex].
2. Calculate Individual Terms:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
1 + 0! = 1 + 1 = 2
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
1 + 1! = 1 + 1 = 2
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
1 + 2! = 1 + 2 = 3
\][/tex]
- For [tex]\(k = 3\)[/tex]:
[tex]\[
1 + 3! = 1 + 6 = 7
\][/tex]
- For [tex]\(k = 4\)[/tex]:
[tex]\[
1 + 4! = 1 + 24 = 25
\][/tex]
3. Sum the Results:
- Now we add up all the calculated terms:
[tex]\[
2 + 2 + 3 + 7 + 25
\][/tex]
- Performing the addition step-by-step:
[tex]\[
2 + 2 = 4
\][/tex]
[tex]\[
4 + 3 = 7
\][/tex]
[tex]\[
7 + 7 = 14
\][/tex]
[tex]\[
14 + 25 = 39
\][/tex]
4. Conclusion:
- The sum of the series [tex]\(\sum_{k=0}^4 (1 + k!)\)[/tex] is:
[tex]\[
\boxed{39}
\][/tex]
