Summation and Infinite Series Flashcard

Front

What is an infinite geometric series?

Back

An infinite geometric series is the sum of the terms of a geometric sequence that continues indefinitely. It converges if the absolute value of the common ratio is less than 1.

Front

How do you find the sum of an infinite geometric series?

Back

The sum S of an infinite geometric series can be calculated using the formula: S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

Front

What is the first term and common ratio in the series 5 + 4 + \frac{16}{5} + \frac{64}{25} + ...?

Back

The first term (a) is 5, and the common ratio (r) is \frac{4}{5}.

Front

Evaluate the infinite series: 5 + \frac{5}{4} + \frac{5}{16} + \frac{5}{64} + ...

Back

The sum is \frac{20}{3}.

Front

What is the formula for the sum of a finite arithmetic series?

Back

The sum S of a finite arithmetic series can be calculated using the formula: S = \frac{n}{2} (a + l), where 'n' is the number of terms, 'a' is the first term, and 'l' is the last term.

Front

What does the notation \sum_{m=3}^{15}(4m+1) represent?

Back

It represents the sum of the expression (4m + 1) for all integer values of m from 3 to 15.

Front

Calculate the sum \sum_{m=3}^{15}(4m+1).

Back

The sum is 481.

Front

What is the formula for the sum of an infinite series of the form \sum_{n=1}^{\infty} ar^{n-1}?

Back

The sum is given by S = \frac{a}{1 - r}, where 'a' is the first term and 'r' is the common ratio.

Front

Evaluate the infinite series: \sum_{x=1}^{\infty} 4096 \left(\frac{1}{4}\right)^{x-1}.

Back

The sum is \frac{16384}{3}.

Front

What is the common ratio in the series 8 + 8 \left(\frac{1}{5}\right) + 8 \left(\frac{1}{5}\right)^{2} + ...?

Back

The common ratio (r) is \frac{1}{5}.