Statement 1 Sr is the sum of an infinite G . P with 1st term r and common ratio (1)/(r+1). Then

step 1 First, just work on statement 2. So, for infinite geometric series, the common ratio is numerica

Statement 1 Sr is the sum of an infinite G . P with 1st term...

Statement 1 $\mathrm{S}_{r}$ is the sum of an infinite $\mathrm{G} . \mathrm{P}$ with 1st term $\mathrm{r}$ and common ratio $\frac{1}{\mathrm{r}+1}$. Then $\mathrm{S}_{\mathrm{r}}-\mathrm{r}$ depends on $\mathrm{r}$. and Statement 2 Sum of an infinite geometric series the common ratio is numerically less than 1 exists only if $\div \mathrm{r} \div<1$

Statement 1 $\mathrm{S}_{r}$ is the sum of an infinite $\mathrm{G} . \mathrm{P}$ with 1st term $\mathrm{r}$ and common ratio $\frac{1}{\mathrm{r}+1}$. Then $\mathrm{S}_{\mathrm{r}}-\mathrm{r}$ depends on $\mathrm{r}$. and
Statement 2 Sum of an infinite geometric series the common ratio is numerically less than 1 exists only if $\div \mathrm{r} \div<1$

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Key Concepts

Infinite Geometric Series

An infinite geometric series is a series in which each term after the first is found by multiplying the preceding term by a constant, called the common ratio. The series has the form a + ar + ar² + ar³ + ... and converges only if the absolute value of the common ratio is less than one.

Convergence Criteria

For an infinite geometric series to converge, the common ratio must satisfy the condition |r| < 1. This ensures that successive terms quickly become very small, allowing the series to sum to a finite number.

Sum Formula for Convergent Series

When an infinite geometric series converges, its sum can be determined using the formula S = a/(1 - r), where 'a' is the first term and 'r' is the common ratio. This formula encapsulates the dependence of the sum on both the initial term and the common ratio.

Parameter Dependency in Series

When elements of a series, such as the first term or the common ratio, are functions of a variable parameter, the resulting sum or expressions derived from the series will depend on that parameter. This reflects how shifts in the parameter influence the behavior and outcome of the series.

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