If |x|<1, then the sum of the infinite series [x+(1)/(2)]+[x^2+x (1)/(2)+((1)/(2))^2]+[x^3+x^2 (

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If |x|<1, then the sum of the infinite series...

If $|\mathrm{x}|<1$, then the sum of the infinite series $\left[\mathrm{x}+\frac{1}{2}\right]+\left[\mathrm{x}^{2}+\mathrm{x} \frac{1}{2}+\left(\frac{1}{2}\right)^{2}\right]+\left[\mathrm{x}^{3}+\mathrm{x}^{2} \frac{1}{2}+\mathrm{x}\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{3}\right]+\ldots . \infty$ (a) $\frac{1}{2}+x$ (b) $\frac{2-x}{1-x}$ (c) $\frac{1+x}{1-x}$ (d) $\frac{2+x}{2-x}$

If $|\mathrm{x}|<1$, then the sum of the infinite series $\left[\mathrm{x}+\frac{1}{2}\right]+\left[\mathrm{x}^{2}+\mathrm{x} \frac{1}{2}+\left(\frac{1}{2}\right)^{2}\right]+\left[\mathrm{x}^{3}+\mathrm{x}^{2} \frac{1}{2}+\mathrm{x}\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{3}\right]+\ldots . \infty$
(a) $\frac{1}{2}+x$
(b) $\frac{2-x}{1-x}$
(c) $\frac{1+x}{1-x}$
(d) $\frac{2+x}{2-x}$

Key Concepts

Algebraic Simplification and Factorization

Once the series is expressed in a summable form, algebraic techniques such as combining like terms, finding common denominators, and factorizing expressions become essential. These operations simplify the result into a compact, closed-form expression, making it easier to interpret and utilize.

Geometric Series Sum Formula

The geometric series sum formula is a powerful tool for summing series where successive terms have a constant ratio. When parts of a complex series can be identified as or reduced to a geometric series, one can directly apply the formula S = a/(1 - r) (for |r| < 1) to obtain the sum, greatly simplifying the analysis.

Convergence of Infinite Series

Understanding when an infinite series converges is fundamental. In many problems, conditions such as |x| < 1 are imposed to ensure that the series does not diverge, particularly when dealing with powers of x. This convergence condition allows the use of infinite series summation formulas and assures that the sum approaches a finite value.

Series Index Shifting

Index shifting is a technique used to modify the starting index of a series to align it with standard summation formulas. By reindexing, complex series expressions can be reformed into more manageable series that are easier to sum, ensuring that all terms are properly accounted for.

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