Question12 Consider the series [math]. Which of the following correctly says what the series is equal to on its interval of convergence?Select one alternative: [math] [math] [math] [math] ResetMaximum marks: 1 Flag question undefined

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BlackTom AI · Accepted Answer

Here is the transcription of the problem and the answer choices as provided, so we can analyze each option carefully.
Question: Question12 Consider the series [math]. Which of the following correctly says what the series is equal to on its interval of convergence? Select one alternative: [math] [math] [math] [math] ResetMaximum marks: 1 Flag question undefined
Answer choices:
- Option 1: ∑_{n=0}^∞ x^n 9^n = 1/(1 - x)
- Option 2: ∑_{n=0}^∞ x^n 9^n = 1/(1 - 9x)
- Option 3: ∑_{n=0}^∞ x^n 9^n = 1/(1 - 9x)
- Option 4: ∑_{n=0}^∞ x^n 9^n = 1/(1 - x^9)

Option-by-option analysis:
- Option 1: ∑_{n=0}^∞ x^n 9^n = 1/(1 - x)
This would only be true if the terms formed a geometric series with ratio x, i.e., sum of (x)^n. However, the given terms multiply x^n by 9^n, which is (9x)^n, not x^n alone. The right-hand side 1/(1 - x) would correspond to ∑ x^n, so this option misidentifies the effective ratio and is incorrect.

- Option 2: ∑_{n=0}^∞ x^n 9^n = 1/(1 - 9x)
Here, notice that x^n 9^n = (9x)^n. If we treat the series as ∑ (9x)^n, it is a geometric series with common ratio 9x. The standard sum formula for a geometric series ∑ r^n from n=0 to ∞ is 1/(1 - r) provided |r| < 1. Substituting r = 9x gives 1/(1 - 9x). This option matches the geometric-series summation correctly, so it is consistent with the series in question.

- Option 3: ∑_{n=0}^∞ x^n 9^n = 1/(1 - 9x)
This option is identical in value to Option 2 and uses the same correct reasoning: since the terms are (9x)^n, the sum is 1/(1 - 9x) for |9x| < 1. There is no discrepancy in the algebra here; it is a proper representation of the sum for the stated series.

- Option 4: ∑_{n=0}^∞ x^n 9^n = 1/(1 - x^9)
This would imply the ratio forms as x^9 per term, which is not the case. The nth term is x^n 9^n = (9x)^n, not x^{9}. The sum 1/(1 - x^9) would correspond to a geometric series with ratio x^9, which does not match the actual ratio 9x. Therefore, this option is incorrect.

Conclusion from the analysis (without naming the final choice): The correct reasoning depends on recognizing that the series is geometric with ratio 9x, so its sum is 1/(1 - 9x) provided the convergence condition |9x| < 1 holds. Options that yield 1/(1 - x) incorrectly treat the ratio as x rather than 9x, and the option with 1/(1 - x^9) misidentifies the ratio as x^9. The two options that present 1/(1 - 9x) are consistent with the (9x)^n form of the terms, while the option presenting 1/(1 - x) is inconsistent with the given terms, and the option with 1/(1 - x^9) is also inconsistent with the (9x)^n ratio.

Question12 Consider the series [math]. Which of the following correctly says what the series is equal to on its interval of convergence?Select one alternative: [math] [math] [math] [math] ResetMaximum marks: 1 Flag question undefined

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