5.3: Series and Convergence | Algebra 2 Honors

How it works

A series adds sequence terms. Finite arithmetic and geometric sums have different formulas. An infinite geometric series has a finite sum only when the common ratio has absolute value less than 1; in that case its terms approach zero fast enough for the partial sums to settle.

S_n=\frac{n(a_1+a_n)}2,\qquad S_\infty=\frac{a_1}{1-r}\quad(|r|<1)

Worked examples

Example 1

Find $\displaystyle 1+2+\cdots+20$ .

Use the arithmetic sum: $\displaystyle S_{20}=20(1+20)/2=210$ .

Example 2

Find the infinite sum $\displaystyle 6+3+3/2+\cdots$ .

The ratio is $\displaystyle 1/2$ , whose absolute value is below $\displaystyle 1$ . Thus $\displaystyle S=6/(1-1/2)=12$ .

Problems

Problem 5.3.1

Find the sum of the first $\displaystyle 10$ terms of $\displaystyle 3,5,7,\ldots$ .

Show worked solution

The tenth term is $\displaystyle 3+9(2)=21$ . Thus $\displaystyle S_{10}=10(3+21)/2=120$ .

Problem 5.3.2

Does $\displaystyle 1+2+4+8+\cdots$ have a finite sum?

Show worked solution

No. The ratio is $\displaystyle 2$ , so $\displaystyle |r|\ge1$ and the infinite geometric series diverges.

Problem 5.3.3

The graph shows partial sums of $\displaystyle 4+2+1+\cdots$ . Find the limiting sum.

The first six partial sums of 4 + 2 + 1 + ... approach the dashed height 8. — The points are partial sums, not the individual series terms.

Show worked solution

The ratio is $\displaystyle 1/2$ , so the infinite sum is $\displaystyle 4/(1-1/2)=8$ . The plotted partial sums approach $\displaystyle 8$ .

The first six partial sums of 4 + 2 + 1 + ... approach the dashed height 8. The infinite geometric sum is 8. — The infinite geometric sum is 8.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 5.3.1

Problem 5.3.2

Problem 5.3.3

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