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Chapter 11: Problem 34

Evaluate each series. $$\sum_{k=1}^{4}(k+1)^{k}$$

Short Answer

Expert verified
700

Step by step solution

01

Understand the series

The given series is o \(\sum_{k=1}^{4}(k+1)^{k}\)o This means the sum of \( (k+1)^k \) for each integer value of \( k \) from 1 to 4.
02

Compute each term in the series

Calculate each term separately:For \( k = 1 \): \((k+1)^k = (1+1)^1 = 2^1 = 2.\)For \( k = 2 \): \((k+1)^k = (2+1)^2 = 3^2 = 9.\)For \( k = 3 \): \((k+1)^k = (3+1)^3 = 4^3 = 64.\)For \( k = 4 \): \((k+1)^k = (4+1)^4 = 5^4 = 625.\)
03

Add the terms

Sum the values obtained:\(2 + 9 + 64 + 625.\)
04

Final sum

Calculate the final sum: \(2 + 9 + 64 + 625 = 700.\)

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Key Concepts

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

summation series
The summation series is a key concept in mathematics, often seen in calculus and algebra. It involves adding up a sequence of numbers. The notation used here, \(\sum_{k=1}^{4}(k+1)^{k}\), represents the sum of a series where each term is defined by a specific formula. In our case, we are summing \((k+1)^k\) from \(k = 1\) to \(k = 4\).

Here are some steps to understand summation series:
  • Identify the range of summation, which is from \(k = 1\) to \(k = 4\) in this case.
  • Understand the formula for each term, \((k+1)^k\), which changes as \(k\) changes.
By breaking down the steps and calculating each term, we can effectively solve the series.
exponents
Exponents are a shorthand way to express repeated multiplication of the same number by itself. In the formula \((k+1)^k\), the exponent \(k\) tells us how many times to multiply \((k+1)\) by itself.

For example:
  • When \(k = 1\), \( (1+1)^1 = 2^1 = 2\).
  • When \(k = 2\), \((2+1)^2 = 3^2 = 9\).
  • When \(k = 3\), \( (3+1)^3 = 4^3 = 64\).
  • When \(k = 4\), \((4+1)^4 = 5^4 = 625\).
Understanding exponents can significantly simplify complex calculations, making it easier to compute terms in our series.
arithmetic operations
Arithmetic operations involve basic calculations such as addition, subtraction, multiplication, and division. In our step-by-step solution, we performed addition to find the sum of calculated terms.

Here鈥檚 a quick rundown of the steps involved:
  • Calculate each term separately using the correct exponent.
  • Sum these terms: \(2 + 9 + 64 + 625\).
  • Finally, perform the addition: 2 + 9 = 11, then 11 + 64 = 75, and finally 75 + 625 = 700.
Accurate arithmetic operations ensure we get the correct final result, which is crucial in evaluating any mathematical series.

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Most popular questions from this chapter

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Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}$$
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