is f(x)=e convergent or divergent?

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24-11-2024

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Is f(x) = eˣ Convergent or Divergent? Understanding Exponential Growth

The question of whether the function f(x) = eˣ is convergent or divergent depends entirely on the context. The function itself doesn't inherently possess convergence or divergence; these concepts apply to series or integrals involving the function, not the function itself. Let's explore this distinction:

1. The Function f(x) = eˣ:

The exponential function f(x) = eˣ is a continuous, smooth function defined for all real numbers. It exhibits exponential growth, meaning its value increases without bound as x increases and approaches zero as x approaches negative infinity. It's not inherently "convergent" or "divergent" in the way a sequence or series would be.

2. Sequences and Series Involving eˣ:

Convergence and divergence are meaningful when discussing sequences or series. Let's consider some examples:

• The Sequence {eⁿ}: This sequence, where the nth term is eⁿ, is clearly divergent. As n approaches infinity, eⁿ approaches infinity. Therefore, the sequence does not have a finite limit.

• The Series Σ e⁻ⁿ (from n=1 to ∞): This is a geometric series with the first term a = e⁻¹ and the common ratio r = e⁻¹. Since |r| = e⁻¹ < 1, this geometric series is convergent. Its sum can be calculated using the formula for the sum of an infinite geometric series: a / (1 - r) = e⁻¹ / (1 - e⁻¹) ≈ 0.58198.

• The Integral ∫₀ˣ eᵗ dt: This definite integral represents the area under the curve of eˣ from 0 to x. As x approaches infinity, the integral also approaches infinity. Thus, the improper integral ∫₀^∞ eᵗ dt is divergent.

• The Integral ∫₋ₓ⁰ eᵗ dt: In contrast to the above, this integral converges to a finite value as x approaches infinity.

In Summary:

The function f(x) = eˣ itself is neither convergent nor divergent. The concepts of convergence and divergence apply to sequences, series, or integrals involving this function. Whether a sequence, series, or integral involving eˣ converges or diverges depends on how eˣ is incorporated and the limits of the summation or integration. To determine convergence or divergence, you must specify the context – the specific sequence, series, or integral being examined. The behavior of eˣ, with its unbounded growth for positive x and approach to zero for negative x, plays a crucial role in determining the convergence or divergence of these mathematical objects.