Question

Find the value of the indicated sum. $$ \sum_{k=1}^{7} \frac{1}{k+1} $$

Short answer

The sum is approximately 1.718.

Step-by-step solution

Write Out the Series Terms

The given sum is \( \sum_{k=1}^{7} \frac{1}{k+1} \). This represents the summation of the terms \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8} \) from \( k=1 \) to \( k=7 \).

Evaluate Each Term

Calculate each term in the series: \( \frac{1}{2} = 0.5 \), \( \frac{1}{3} \approx 0.333 \), \( \frac{1}{4} = 0.25 \), \( \frac{1}{5} = 0.2 \), \( \frac{1}{6} \approx 0.167 \), \( \frac{1}{7} \approx 0.143 \), \( \frac{1}{8} = 0.125 \).

Sum the Terms Together

Add up the values we've calculated: \(0.5 + 0.333 + 0.25 + 0.2 + 0.167 + 0.143 + 0.125 \approx 1.718 \).

State the Final Answer

After adding all the terms, the sum of the series is \( \approx 1.718 \).

Key concepts

Series

In the world of calculus and mathematical sequences, the concept of a series is fundamental. A series is essentially the sum of a sequence of numbers.
When these numbers follow a specific pattern or rule, the sequence can be neatly expressed in a formula.
Such as the summation notation we often see, which uses the Greek letter Sigma \((Σ)\). In the exercise provided, we see the terms summed up as: \(\sum_{k=1}^{7} \frac{1}{k+1}\). This means we take each integer from 1 to 7, plug it into the formula \(\frac{1}{k+1}\), and sum them all together.

• Series: Finite set of elements that are summed.
• Pattern: Follows a specific rule or formula.
• Notation: Often uses Sigma (Σ) for concise representation.

Recognizing a pattern in a series helps in understanding the overall structure and knowing how to compute them easily and efficiently.

Summation

Summation is a process in mathematics where we add together all the terms in a series. The notation \(\sum\) signifies this operation and helps us sum multiple terms in a compact form.
For example, in the exercise, \(\sum_{k=1}^{7} \frac{1}{k+1}\) describes adding up several fractions, beginning from \( k=1 \) to \( k=7 \).
This allows us to represent the addition of multiple terms succinctly, which is especially helpful in lengthy calculations. The summation notation tells us exactly which terms to add and ensures that we follow a systematic approach.

• Notation: \(Σ\) indicates summation.
• Range: Specified by the limits of the summation.
• Application: Used to condense the addition of many terms.

Learning summation is crucial because it is a stepping stone to more complex mathematical concepts, and it simplifies the addition of numerous values.

Arithmetic Operations

Arithmetic operations, such as addition and division performed in the context of calculus, are essential skills. In the given problem, we're primarily dealing with division when evaluating each fractional term, followed by addition to find the sum.
Evaluating each term in a series involves dividing one number by another to determine the value of each individual fraction.
Afterward, we combine these results through addition. Ensuring accurate arithmetic when performing these operations is critical to obtaining the right answer.

• Division: Calculate each fractional term.
• Addition: Combine the evaluated terms.
• Calculations: Must be precise for correct results.

Being adept in these basic arithmetic operations not only helps in solving series but is also crucial for a broad range of mathematical applications.

Fraction

Fractions represent parts of a whole and are a significant part of the arithmetic operations used in solving series problems. In our exercise, each term in the series is a fraction like \(\frac{1}{k+1}\).
This requires us to understand how fractions are used and calculated.
The key is in identifying the numerator and the denominator and knowing how to convert them into decimal form if necessary, which makes summing them simpler.

• Numerator/Denominator: Parts of a fraction.
• Conversion: Fractions can be converted into decimals.
• Usage: Essential in calculus for representing parts of a series.

The ability to manipulate fractions is vital, especially when dealing with series, as it ensures accurate computation of each individual term.