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Chapter 12: Problem 36

Find the sum for each series. $$\sum_{i=1}^{5} i^{i-1}$$

Short Answer

Expert verified
The sum of the series is 701.

Step by step solution

01

Understand the Series

The expression \( \sum_{i=1}^{5} i^{i-1} \) represents the sum of terms where each term is calculated as \( i^{i-1} \) for \( i \) ranging from 1 to 5.
02

Calculate Each Term Individually

Compute each term in the series:- For \( i = 1 \), the term is \( 1^{1-1} = 1^0 = 1 \).- For \( i = 2 \), the term is \( 2^{2-1} = 2^1 = 2 \).- For \( i = 3 \), the term is \( 3^{3-1} = 3^2 = 9 \).- For \( i = 4 \), the term is \( 4^{4-1} = 4^3 = 64 \).- For \( i = 5 \), the term is \( 5^{5-1} = 5^4 = 625 \).
03

Summing Up the Terms

Add all the calculated terms together to find the sum of the series:\[ 1 + 2 + 9 + 64 + 625 = 701 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Powers and Exponents
In mathematics, powers and exponents are a way to express repeated multiplication of the same number by itself. The concept is crucial for simplifying large expressions and for many mathematical operations. For example, in the exercise, each term of the series is expressed using exponents. Here鈥檚 how it breaks down:
  • The expression \( i^{i-1} \) is based on the concept of powers.
  • In this scenario, \( i \) is the base, and \( i-1 \) is the exponent.
  • It means you multiply the base \( i \) , \( (i-1) \) times by itself.

Understanding this helps simplify calculations. If you encounter \( 2^3 \), you simply multiply 2 by itself three times, i.e., \( 2 \times 2 \times 2 = 8 \). The practice of powers helps manage expressions that would otherwise be unwieldy if expressed as repeated multiplication.
Summation Notation
Summation notation, represented by the Greek letter sigma (\( \Sigma \)), provides a concise way to express sums. This notation is particularly useful for summing series of numbers or terms that follow a specific pattern.
In our exercise, \( \sum_{i=1}^{5} i^{i-1} \) demonstrates its efficiency:
  • \( \Sigma \) indicates the operation of addition.
  • \( i=1 \) is the starting index, denoting where the summation starts.
  • 5 is the upper limit of the index, showing where it ends.
  • Each term \( i^{i-1} \) is computed within the limits, then summed together.

This notation allows mathematicians and students to handle complex series without writing out every individual calculation, making it an invaluable tool in mathematics.
Arithmetic Calculations
Arithmetic calculations involve the basics of addition, subtraction, multiplication, and division鈥攁ll foundational operations in mathematics. In solving series sums, such as our exercise, comprehension of these calculations is essential:
  • Each term鈥檚 value is calculated using exponents.
  • The results of these calculations are then added together to find the total sum.

For instance, after calculating each term \( (1, 2, 9, 64, 625) \), the next step involves adding them:
  • Start by lining them up: \(1 + 2 = 3\)
  • Then add the next: \(3 + 9 = 12\)
  • Continue: \(12 + 64 = 76\)
  • And finally: \(76 + 625 = 701\)

Following these sequential steps ensures accuracy in solving the problem, reinforcing the importance of arithmetic skills.

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Most popular questions from this chapter

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, 1)=n$$

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, n-r)=C(n, r)$$

Use a formula to find the sum of each arithmetic series. $$7.5+6+4.5+3+1.5+0+(-1.5)$$

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{k=1}^{n} 2 k \leq 42$$

Prove each statement by mathematical induction. \((a b)^{n}=a^{n} b^{n}\) (Assume that \(a\) and \(b\) are constant.)
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