Infinite series

1003108709

Level:

We are given an infinite geometric series:

\sum_{n = 1}^{\infty} \frac{\sqrt{3}}{2^{n - 1}} .

What is the value of its first term

a_{1}

\sqrt{3}

\frac{\sqrt{3}}{2}

\frac{\sqrt{3}}{4}

\frac{1}{2}

3

1003108708

Level:

We are given an infinite geometric series:

\sum_{n = 1}^{\infty} {(- \frac{1}{2})}^{n - 1} .

What is the value of its second term

a_{2}

- \frac{1}{2}

1

\frac{1}{2}

\frac{1}{4}

- 1

1003108707

Level:

We are given an infinite geometric series:

(\sqrt{5} - \sqrt{3}) + (5 - \sqrt{15}) + (5 \sqrt{5} - 5 \sqrt{3}) + \dots .

What is the value of its common ratio?

\sqrt{5}

\sqrt{5} - \sqrt{3}

\sqrt{5} - \sqrt{3} + 5

\sqrt{5} + 5

5

1003108706

Level:

We are given an infinite geometric series:

\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - \frac{16}{81} + \dots .

What is the value of its common ratio?

- \frac{2}{3}

\frac{2}{3}

\frac{2}{9}

- \frac{2}{9}

\frac{4}{27}

1003108705

Level:

We are given an infinite geometric series:

4 + \frac{8}{3} + \frac{16}{9} + \frac{32}{27} + \dots .

What is the value of its common ratio?

\frac{2}{3}

\frac{1}{3}

\frac{4}{3}

\frac{3}{2}

\frac{3}{4}

1003108704

Level:

Expand the given sum by listing individual terms.

\sum_{n = 2}^{6} (4 n - 5)

3 + 7 + 11 + 15 + 19

- 1 + 3 + 7 + 11 + 15

4 + 8 + 12 + 16 + 20

3 + 8 + 11 + 15 + 19

8 - 5 + 12 + 16 + 20

1003108703

Level:

Use summation notation to express the given infinite series.

\frac{3}{x^{3}} + \frac{3}{x^{2}} + \frac{3}{x} + 3 + 3 x + \dots

\sum_{n = 1}^{\infty} 3 \cdot x^{n - 4}

\sum_{n = 1}^{\infty} 3 \cdot x^{n - 3}

\sum_{n = 1}^{\infty} 3 \cdot x^{n + 3}

\sum_{n = 1}^{\infty} 3 \cdot x^{n + 4}

\sum_{n = 1}^{\infty} 3 \cdot x^{n}

1003108702

Level:

Use summation notation to express the given infinite series.

- 1 + 2 - 4 + 8 - 16 + \dots

\sum_{n = 1}^{\infty} (- 1)^{n} \cdot 2^{n - 1}

\sum_{n = 1}^{\infty} (- 1)^{n - 1} \cdot 2^{n - 1}

\sum_{n = 1}^{\infty} (- 1)^{n + 1} \cdot 2^{n - 1}

\sum_{n = 1}^{\infty} (- 1)^{n} \cdot 2^{n + 1}

\sum_{n = 1}^{\infty} (- 1)^{n} \cdot 2^{n}

1003108701

Level:

Use summation notation to express the given infinite series.

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots

\sum_{n = 1}^{\infty} \frac{1}{2^{n - 1}}

\sum_{n = 1}^{\infty} \frac{1}{2^{n}}

\sum_{n = 1}^{\infty} \frac{1}{2^{n + 1}}

\sum_{n = 1}^{\infty} \frac{1}{2^{2 n}}

\sum_{n = 1}^{\infty} \frac{1}{2^{2 n - 1}}

9000073407

Level:

Find all the values of

x

such that the following infinite series is convergent.

1 + 3 - 2 x + (3 - 2 x)^{2} + (3 - 2 x)^{3} + \dots

x \in (1; 2)

x \in (- \infty; - 1)

x \in (1; + \infty)

x \in R

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