Problem 30 - Calculus for Biology and Medicine

Question

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=0}^{10}(-1)^{k} $$

Step-by-Step Solution

Answer

The sum is 1.

Step 1: Identify the Nature of the Sum

The sum given is \( \sum_{k=0}^{10}(-1)^{k} \). Notice that the terms alternate between 1 and -1 based on whether \( k \) is even or odd.

Step 2: Evaluate the Individual Terms

To see the pattern, calculate the first few terms: when \( k = 0 \), \((-1)^0 = 1\); when \( k = 1 \), \((-1)^1 = -1\); and it continues this way, alternating.

Step 3: Count Positive and Negative Terms

Since \( k \) ranges from 0 to 10, there are 11 terms in total. When counted, there will be 6 positive \( 1's \) (for even \( k \)) and 5 negative \( -1's \) (for odd \( k \)).

Step 4: Calculate the Sum

The sum of the 6 positive terms is \( 6 \times 1 = 6 \) and the sum of the 5 negative terms is \( 5 \times -1 = -5 \). The total sum is \( 6 - 5 = 1 \).

Key Concepts

Summation Formula

A summation formula is a simplified equation used to calculate the sum of a sequence of numbers. It often saves us time compared to adding up numbers individually. One of the most famous summation formulas is the sum of the first "n" natural numbers, given by:

• \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \)

This allows us to calculate quickly how many total candies, for example, we would have if we lined them up in a perfect sequence from 1 to "n." There is also a handy formula for the sum of the squares of the first "n" numbers:

• \( \sum_{k=1}^{n} k^{2} = \frac{n(n+1)(2n+1)}{6} \)

Understanding these formulas and knowing when to use them can greatly simplify many algebra problems. When using summation formulas, always make sure to identify the range and the pattern of the terms to apply the correct formula.

Alternating Series

An alternating series is a sequence where the sign of each term alternates between positive and negative. This alternation of signs introduces a pattern that can simplify finding the sum of the series, especially when it is a finite series. Take for instance the series \( \sum_{k=0}^{10} (-1)^{k} \), where the sign of each term alternates based on whether \( k \) is even or odd. This pattern can significantly affect the overall sum:

• If \( k \) is even, \((-1)^k = 1\), contributing a positive term,
• If \( k \) is odd, \((-1)^k = -1\), contributing a negative term.

Alternating series such as this can be solved by carefully counting the number of positive and negative contributions. Recognizing this pattern allows for the quick calculation of sums without needing to add each term individually.

Finite Series

A finite series is a series that has a fixed number of terms. This contrasts with an infinite series, which continues indefinitely. Understanding finite series is key to mastering sums in algebra, as they allow us to limit our calculations to a manageable number of terms. In our exercise, the series \( \sum_{k=0}^{10} (-1)^{k} \) is considered finite because it starts at \( k = 0 \) and ends at \( k = 10 \), yielding exactly 11 terms.

• To evaluate this finite series, observe the range of terms and the pattern present.
• Examine the contributions by counting how many terms are positive and how many are negative.

Through this observation, one can determine that the sum of the series is simply the difference between the total positive and negative terms. Recognizing series as finite allows us to apply specific strategies for efficient and accurate summation.