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An 2 5 N As N Go to Infinity Calculator

Reviewed by Calculator Editorial Team

This calculator determines the limit of the sequence an = 2^(5n) as n approaches infinity. The sequence grows without bound as n increases, and we'll show you how to calculate this mathematically.

What is the limit of a sequence?

The limit of a sequence describes its behavior as the index n approaches infinity. For the sequence an = 2^(5n), we're interested in what happens to the terms as n becomes very large.

In this case, the sequence grows exponentially because the exponent 5n increases without bound as n increases. This means the terms of the sequence become extremely large as n approaches infinity.

How to calculate the limit of an = 2^(5n)

To find the limit of an = 2^(5n) as n → ∞, we can use the properties of exponential functions:

lim (n→∞) 2^(5n) = ∞

This is because the exponent 5n grows without bound as n increases, causing the entire expression to approach infinity.

The general rule is that for any positive real number a > 1 and any real number b, the limit of a^(bn) as n → ∞ is infinity if b > 0.

Worked example

Let's calculate the limit of an = 2^(5n) as n → ∞ step by step:

  1. Identify the sequence: an = 2^(5n)
  2. Recognize that the exponent 5n grows without bound as n increases
  3. Apply the limit property: lim (n→∞) a^(bn) = ∞ when a > 1 and b > 0
  4. Conclusion: The limit is infinity

This shows that as n becomes very large, the terms of the sequence become extremely large without bound.

FAQ

What does it mean when a sequence has an infinite limit?
An infinite limit means the sequence grows without bound as n increases. The terms become larger and larger without ever reaching a finite value.
Can a sequence have a finite limit?
Yes, sequences can have finite limits. For example, the sequence an = 1/n approaches 0 as n → ∞.
What's the difference between sequence limits and function limits?
Sequence limits describe the behavior of terms in a sequence as the index approaches infinity, while function limits describe the behavior of a function as its input approaches a certain value.