Given geometrie ser ies $$ 9 x^{2}+6 x^{3}+4 x^{4} $$ for whien values of $$ x $$ will selles converge

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To determine the values of $$ x $$ for which the series $$ 9x^2 + 6x^3 + 4x^4 $$ converges, we first need to recognize that this is a polynomial expression rather than a traditional infinite series. However, we can analyze the convergence of the series by considering it as a power series.

The general form of a power series is:

$$
\sum_{n=0}^{\infty} a_n x^n
$$

In this case, we can rewrite the given expression as:

$$
\sum_{n=2}^{4} a_n x^n = 9x^2 + 6x^3 + 4x^4
$$

Since this is a finite polynomial, it converges for all values of $$ x $$.

However, if we were to consider an infinite series that resembles this polynomial, we would typically look for a ratio or root test to determine convergence.

For a geometric series, the series converges if the absolute value of the common ratio $$ r $$ is less than 1:

$$
|r| < 1
$$

In this case, since we are dealing with a polynomial, we conclude that the series converges for all $$ x $$.

Thus, the final answer is:

The series converges for all values of $$ x $$.
The series converges for all values of $$ x $$.

Given geometrie ser ies \( 9 x^{2}+6 x^{3}+4 x^{4} \) for whien values of \( x \) will selles converge

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