For what values of x(x≠0) will the following infinite geometric series have a finate sum?
x+3x²+9x³+...
3x²/x
what are the next steps?
Thanks in advance.
The ratio of succesasive terms is 3x. This must be less than 1 for the infinite series to have a finite sum. Therefore x < 1/3 is required.
Thanks!
To determine for what values of x (x ≠ 0) the infinite geometric series has a finite sum, we need to find the common ratio of the series.
From the given series: x + 3x² + 9x³ + ...
Notice that each term, starting from the second term, can be obtained by multiplying the previous term by the common ratio. Therefore, we can find the common ratio by dividing any term by its preceding term.
Let's take the second term and divide it by the first term:
(3x²) / x = 3x
We have found that the common ratio of the series is 3x.
For a geometric series to have a finite sum, the common ratio must be between -1 and 1 (excluding -1 and 1 themselves). Therefore, we can set up an inequality to find the range of x values:
-1 < 3x < 1
Next, we divide all parts of the inequality by 3 to isolate x:
-1/3 < x < 1/3
So, the values of x (x ≠ 0) for which the given infinite geometric series has a finite sum are -1/3 < x < 1/3.
As for the "3x²/x" expression you provided, it seems to be the second term divided by the first term. However, this doesn't directly relate to determining the range of x values for the series to have a finite sum. The common ratio, determined as explained above, is what we need to focus on to solve this problem.