Evaluate the Summation sum from i=1 to 5 of ((-1)^i)/((i+1)!)
The question is asking for the evaluation of a finite mathematical series. In the series, each term is formed by a fraction with a numerator that alternates in sign depending on the value of 'i' (the index of summation) because of the factor (-1)^i. The denominator of each fraction is (i+1) factorial, noted as (i+1)!. The factorial of a number n, denoted n!, is the product of all positive integers less than or equal to n. The summation process involves adding these terms together in sequence, starting with i=1 and ending with i=5. The result should yield the total sum of this series after all terms are added up according to these rules.
$\sum_{i = 1}^{5} \frac{\left(\left(\right. - 1 \left.\right)\right)^{i}}{\left(\right. i + 1 \left.\right) !}$
Answer
Solution:
Step:1 Write out the terms of the series for each integer value of $i$ from 1 to 5.
$$\frac{(-1)^1}{(1+1)!} + \frac{(-1)^2}{(2+1)!} + \frac{(-1)^3}{(3+1)!} + \frac{(-1)^4}{(4+1)!} + \frac{(-1)^5}{(5+1)!}$$
Step:2 Calculate the sum of the series.
$$- \frac{53}{144}$$
Step:3 Express the final sum in different formats.
Exact Form: $$- \frac{53}{144}$$ Decimal Form: $$-0.3681$$
Solution:"The sum of the series from i=1 to 5 of $((-1)^i)/((i+1)!)$ is $- \frac{53}{144}$ or approximately $-0.3681$ in decimal form."
Knowledge Notes:
The problem involves evaluating a finite series where each term is defined by a function of $i$. The function involves an alternating sign, which is achieved by raising $-1$ to the power of $i$, and a factorial in the denominator, which is a product of all positive integers up to a certain number (denoted by $n!$).
Key points to understand in solving this problem include:
Factorial Notation: For any positive integer $n$, $n!$ (n factorial) is the product of all positive integers less than or equal to $n$. For example, $4! = 4 \times 3 \times 2 \times 1 = 24$.
Series Expansion: To evaluate the sum, each term of the series must be expanded based on the given function of $i$.
Alternating Series: An alternating series is one in which the signs of the terms alternate between positive and negative. In this case, the sign alternates because of the $(-1)^i$ factor.
Simplification: After expanding the series, the terms are simplified and added together to find the sum.
Representation of Results: The result can be presented in an exact form (as a fraction) or in a decimal form for easier interpretation.
