C-07 — Infinite Series

When some numbers are arranged successively under a definite rule such that the relationship between any two successive terms is known, the resulting set is called a —

For the set of natural numbers $\{1, 2, 3, 4, \ldots\}$, the corresponding set of square numbers is obtained by the rule $f(n) =$ ?

The general term of the sequence $1, 4, 9, 16, \ldots$ is —

Standard notation for a sequence with general term $n^2$ is —

In the sequence $1, 4, 9, 16, \ldots$, the second term is —

The number of terms of any sequence is —

The general term of the sequence $1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \ldots$ is —

The general term of the sequence $3, 1, -1, -3, \ldots$ is —

The general term of the sequence $\dfrac{1}{2}, \dfrac{1}{5}, \dfrac{1}{10}, \dfrac{1}{17}, \ldots$ is —

The general term of the sequence $1, 3, 5, 7, \ldots$ is —

The general term of the sequence $1, \sqrt{2}, \sqrt{3}, 2, \ldots$ is —

The sequence generated by the general term $1 + (-1)^n$ is —

The sequence generated by the general term $1 - (-1)^n$ is —

The first term of the sequence with general term $\dfrac{n}{2n - 1}$ is —

If the terms of a sequence are connected successively by a "$+$" sign, then the result is called a —

A series in which the ratio between two successive terms is constant is called —

A series in which the difference between two consecutive terms is constant is called —

The series $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \ldots$ is an example of a —

Depending on the number of terms, series can be divided into how many classes?

Which of the following is one of the two classes of series based on number of terms?

Consider the following statements:

If $u_1, u_2, u_3, \ldots, u_n, \ldots$ is a sequence of real numbers, then $u_1 + u_2 + u_3 + \ldots + u_n + \ldots$ is called —

In the infinite series $u_1 + u_2 + u_3 + \ldots + u_n + \ldots$, $u_n$ is called the —

The $n$-th partial sum $S_n$ of an infinite series is the sum of —

For an infinite series, the 3rd partial sum $S_3$ is —

The sum of the first $n$ terms of the arithmetic series $1 + 2 + 3 + 4 + \ldots$ is —

The value of $S_{10}$ for the series $1 + 2 + 3 + 4 + \ldots$ is —

The value of $S_{100}$ for the series $1 + 2 + 3 + \ldots$ is —

The value of $S_{1000}$ for the series $1 + 2 + 3 + \ldots$ is —

As $n$ increases, the partial sum $S_n$ of $1 + 2 + 3 + \ldots$ —

The infinite series $1 + 2 + 3 + 4 + \ldots$ —

For the series $1 - 1 + 1 - 1 + \ldots$, the second partial sum $S_2$ is —

For the series $1 - 1 + 1 - 1 + \ldots$, the third partial sum $S_3$ is —

For the series $1 - 1 + 1 - 1 + \ldots$, when $n$ is even, $S_n$ equals —

For the series $1 - 1 + 1 - 1 + \ldots$, when $n$ is odd, $S_n$ equals —

The sum of the infinite series $1 - 1 + 1 - 1 + \ldots$ —

The $n$-th term of the geometric series $a + ar + ar^2 + ar^3 + \ldots$ is —

For the geometric series $a + ar + ar^2 + \ldots$ with $r \ne 1$, the partial sum up to $n$-th term is —

The sum of the infinite geometric series $a + ar + ar^2 + \ldots$ exists when —

When $|r| < 1$ and $n \to \infty$, the value of $|r^n|$ —

If $|r| < 1$, the sum to infinity of $a + ar + ar^2 + \ldots$ is —

The notation for the sum to infinity of an infinite geometric series is —

If $r = 1$, the infinite geometric series becomes $a + a + a + \ldots$ and its $n$-th partial sum is —

If $r = -1$, the infinite geometric series is $a - a + a - a + \ldots$, whose sum —

If $|r| > 1$ in an infinite geometric series, then —

The condition $|r| < 1$ for $r$ is equivalent to —

For the infinite geometric series with $a = 4$, $r = \dfrac{1}{2}$, the sum to infinity is —

For the infinite geometric series with $a = 2$, $r = \dfrac{1}{3}$, the sum to infinity is —

For the infinite geometric series with $a = 3$, $r = 3$, the sum to infinity —

For the infinite geometric series with $a = 5$, $r = \dfrac{1}{2}$, the sum to infinity is —

For the infinite geometric series with $a = 81$, $r = -\dfrac{1}{3}$, the sum to infinity is —

The sum to infinity of $1 + 0.1 + 0.01 + 0.001 + \ldots$ is —

For the infinite geometric series $1 + \dfrac{1}{\sqrt{2}} + \dfrac{1}{2} + \dfrac{1}{2\sqrt{2}} + \ldots$, the sum to infinity is —

Consider the following statements about an infinite geometric series $a + ar + ar^2 + \ldots$:

As an infinite geometric series, $0.\overline{5}$ is written as —

The common ratio of the geometric series $0.5 + 0.05 + 0.005 + \ldots$ is —

The rational fraction representing $0.\overline{5}$ is —

The rational fraction representing $0.\overline{12}$ is —

For converting $1.\overline{231}$ to a fraction, the infinite geometric part has first term and common ratio —

The rational fraction representing $1.\overline{231}$ is —

For the infinite geometric series $\dfrac{1}{2x+1} + \dfrac{1}{(2x+1)^2} + \dfrac{1}{(2x+1)^3} + \ldots$, when $x = 1$, the common ratio is —

For the same series with $x = \dfrac{3}{2}$, the first term is —

For the same series with $x = \dfrac{3}{2}$, the 5th term is —

For the same series with $x = \dfrac{3}{2}$, the sum of first $10$ terms is —

For the series $\dfrac{1}{2x+1} + \dfrac{1}{(2x+1)^2} + \ldots$ to have sum to infinity, the condition on $x$ is —

For the same series, the sum to infinity (when it exists) is —

What is the 12th term of the series $1, 3, 5, 7, \ldots$?

What is the 3rd term of a sequence whose $n$-th term is $\dfrac{1}{n(n+1)}$?

If the $n$-th term of a sequence is $u_n = 1 - (-1)^n$, then —

The 10th term of the sequence $2, 4, 6, 8, 10, 12, \ldots$ is —

The 15th term of the sequence $2, 4, 6, 8, 10, \ldots$ is —

The $r$-th term of the sequence $2, 4, 6, 8, \ldots$ is —

The 10th term of the sequence whose $n$-th term is $\dfrac{1}{n(n+1)}$ is —

The 15th term of the sequence whose $n$-th term is $\dfrac{1}{n(n+1)}$ is —

The 10th term of the sequence $0, 1, 0, 1, 0, 1, \ldots$ is —

The 15th term of the sequence $0, 1, 0, 1, 0, 1, \ldots$ is —

For the sequence with $u_n = \dfrac{1}{n}$, if $u_n < 10^{-8}$, the value of $n$ satisfies —

For the sequence with $u_n = \dfrac{1}{n}$, if $u_n > 10^{-8}$, the value of $n$ satisfies —

For the sequence with $u_n = \dfrac{1}{n}$, when $n$ is sufficiently large, the limiting value of $u_n$ is —

The sum to infinity of $1 + \dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{27} + \ldots$ is —

The sum to infinity of $8 + 4 + 2 + 1 + \dfrac{1}{2} + \ldots$ is —

The sum to infinity of $1 + 2 + 4 + 8 + 16 + \ldots$ —

The sum of the first $n$ terms of the series $7 + 77 + 777 + \ldots$ is —

The sum of the first $n$ terms of the series $5 + 55 + 555 + \ldots$ is —

For the series $\dfrac{1}{x+1} + \dfrac{1}{(x+1)^2} + \dfrac{1}{(x+1)^3} + \ldots$, the sum to infinity exists when —

For the same series, the sum to infinity (when it exists) is —

The repeating decimal $0.\overline{27}$ as a rational fraction is —

The repeating decimal $0.\overline{0123}$ as a rational fraction is —

The sum of three consecutive terms of a geometric series is $24$ and their product is $64$. Taking the three terms as $\dfrac{a}{r}, a, ar$, the equation from the product is —

From the same problem, the value of $a$ is —

From the same problem, if the common ratio is $\dfrac{1}{2}$, then the first term of the three is —

From the same problem, with first term $8$ and common ratio $\dfrac{1}{2}$, the sum of the resulting infinite geometric series is —

What is the 20th term of a sequence whose $n$-th term is $(-1)^n$?

The $n$-th term of a sequence is $u_n = \dfrac{1}{n}$ and $u_n < 10^{-4}$. Then —

What is the 10th term of the series $4 + \dfrac{4}{3} + \dfrac{4}{9} + \ldots$?