How do you find the actual limit after knowing the sequence converges?

Conditions

• The sequence is known to converge to some real number $L$.
• The recurrence relation is $a_{n+1}=\frac{1}{3}(a_n+1)$.

Reasoning, step by step

1. Write the recurrence relation: $a_{n+1}=\frac{1}{3}(a_n+1)$.
2. Take the limit as $n \to \infty$ on both sides: $\lim_{n\to\infty} a_{n+1} = \lim_{n\to\infty} \frac{1}{3}(a_n+1)$.
3. Substitute $L$ for the limits of $a_{n+1}$ and $a_n$: $L = \frac{1}{3}(L+1)$.
4. Multiply by 3: $3L = L + 1$.
5. Subtract $L$: $2L = 1$.
6. Divide by 2: $L = \frac{1}{2}$.

Example

The board shows the calculation: $L=\frac{1}{3}(L+1)\Rightarrow \frac{2}{3}L=\frac{1}{3}\Rightarrow L=\frac{1}{2}$.

Common misconceptions

• Assuming $a_{n+1}$ converges to a different limit than $a_n$.
• Forgetting to apply the limit to the constant term inside the parentheses.

Watch the explanation

Explore next

• What is the limit of the sequence $a_{n+1}=\frac{1}{3}(a_n+1)$ with $a_1=1$?
• Why must the sequence be bounded and monotone before taking the limit?