Problem 98 - Precalculus with Limits

Question

In Exercises 97-102, use a calculator to find the sum. \( \displaystyle \sum_{j=1}^{10}\frac{3}{j + 1} \)

Step-by-Step Solution

Answer

To find out the final answer, one needs to add together each of these categories. The calculator can help obtain the precise decimal or fractional representation.

Step 1: Identify the Summation Terms

Inside the summation symbol we have the fraction \( \frac{3}{j + 1} \), with j ranging from 1 to 10. This indicates that the summation will include 10 total terms, where j is replaced by successive integers from 1 through 10.

Step 2: Compute Each Term

Substitute every value from 1 to 10 for j gradually, compute the fraction for each term using a calculator. These are the computed terms: \( \frac{3}{2}, \frac{3}{3}, \frac{3}{4}, \frac{3}{5}, \frac{3}{6}, \frac{3}{7}, \frac{3}{8}, \frac{3}{9}, \frac{3}{10}, \frac{3}{11} \).

Step 3: Summation

Add all of the computed terms using a calculator. The sum of the terms will be the result of the summation.

Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. In an arithmetic sequence, each term increases by a constant difference. For example, the sequence 1, 2, 3, 4 is arithmetic because each term increases by 1.

However, the original exercise deals with a sequence where each term requires division and isn't in the simplest arithmetic form. But understanding regular arithmetic series helps us realize that even complex forms have regular patterns. In our specific problem, the terms are structured around fractions: \( \frac{3}{j+1} \).

Here, we aren't strictly dealing with an arithmetic series in the classic sense because the increment involves dividing 3 by an ever-increasing denominator \(j+1\). However, the computation becomes simpler if we think of it as summing individual terms that follow a consistent pattern as \(j\) increases.

Partial Sums

Partial sums in a series involve finding the sum of the first few terms of the series. This helps understand how a series converges or behaves. It also helps verify that calculations are on track, especially in lengthy series.

In the original step-by-step solution, calculating each term individually provides partial sums. For example, finding the sum of the first five fractions:

• \(\frac{3}{2} + \frac{3}{3} + \frac{3}{4} + \frac{3}{5} + \frac{3}{6}\)

These partial sums offer insight and accuracy-checking even before reaching the final sum. The advantage of this process is catching mistakes earlier and gaining a deeper understanding of how small parts contribute to the total sum.

Finite Series

A finite series is a sequence of numbers that has an end. Unlike infinite series that continue endlessly, finite series have a set number of terms. In the given exercise, we work with a finite series where \(j\) ranges from 1 to 10.
Thus, the series sums 10 specific fractions terms starting with \(\frac{3}{2}\) and ending with \(\frac{3}{11}\).

This characteristic ensures the summation process concludes and provides a definite result. Working with a finite series, it’s easier because upon substituting each value of \(j\), you can immediately calculate, verify and conclude the sum systematically.

The structured approach minimizes errors and facilitates the process, allowing calculations to be performed with certainty using tools like calculators.