Determine whether the sequence converges or diverges. If it converges, find the limit. If necessary, use L'H rule.

Question

a_n=ⁿ√(2¹⁺³ⁿ⁾)

In general, with roots and exponents, do you take the natural log in order to find the limit?

Answer 1

To determine whether the sequence a_n = ⁿ√(2⁽¹⁺³ⁿ⁾) converges or diverges, let's analyze its behavior as n approaches infinity.

First, rewrite the expression:

a_n = (2^(1+3n)/n) = 2^(1/n+3)

Now, as n approaches infinity, the exponent 1/n approaches 0. So, the expression becomes:

a_n = 2⁽⁰⁺³⁾ = 8

Therefore, the limit of the sequence is lim_n→∞ a_n = 8.

Hence, the sequence converges, and its limit is 8.

When looking for a limit, there are 9 main operations you can perform.

1. Direct Substitution: Substitute the value of n into the expression and evaluate it directly, unless it results in an undefined expression or leads to an indeterminate form.
2. Algebraic Manipulation: Simplify the expression algebraically by factoring, rationalizing the numerator or denominator, combining like terms, or using algebraic identities.
3. Squeeze Theorem: If you can find two functions that bound the given function and have the same limit at the point of interest, then the given function's limit must also be that value.
4. Properties of Limits: Utilize properties such as the limit laws, which include properties like the sum/difference law, product law, quotient law, and power law of limits.
5. Special Limits: Use known limits for basic functions, such as trigonometric, exponential, and logarithmic functions.
6. L'Hôpital's Rule: When you have an indeterminate form like 0/0​ or ∞/∞​, you can sometimes use L'Hôpital's Rule, which involves taking the derivatives of the numerator and denominator and then re-evaluating the limit.
7. Rationalization: For expressions involving radicals, rationalize the expression by multiplying by the conjugate to eliminate the radical in the denominator. Note that this is like multiplying by 1
8. Trigonometric Identities: Utilize trigonometric identities to simplify trigonometric expressions.
9. Sums and Products: Splitting a limit into the sum or product of two or more limits if possible.