Problem 36

Question

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

Step-by-Step Solution

Verified

Answer

The sum of the series is 701.

1Step 1: 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.

2Step 2: 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 $.

3Step 3: Summing Up the Terms

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

Key Concepts

Powers and ExponentsSummation NotationArithmetic Calculations

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’s 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—all foundational operations in mathematics. In solving series sums, such as our exercise, comprehension of these calculations is essential: