
lim
⁡
𝑛
→
∞
∑
𝑖
=
1
𝑛
1
𝑛
(
1
+
2
𝑖
𝑛
)
10
n→∞
lim
​
  
i=1
∑
n
​
  
n(1+ 
n
2i
​
 ) 
10
 
1
​


It looks like you're dealing with a sum and a limit as \( n \to \infty \). Let me clarify and restate the expression you provided:

\[
\lim_{n \to \infty} \sum_{i=1}^{n} \frac{1}{n} \left( 1 + \frac{2i}{n} \right)^{10}
\]

This type of limit involving a sum is often handled by converting it into an integral as \( n \to \infty \), where the sum resembles a Riemann sum. Here's how we can approach this:

### Step 1: Recognize the Riemann Sum
The given sum looks like a Riemann sum, where:
- \( \frac{1}{n} \) corresponds to \( \Delta x \) (the width of small intervals).
- \( 1 + \frac{2i}{n} \) is a function evaluated at points \( \frac{i}{n} \), which tends to \( x \) as \( n \to \infty \).

Thus, we can rewrite the sum as an approximation of an integral. The limits of the sum, \( i = 1 \) to \( n \), correspond to the interval \( x = 1 \) to \( x = 3 \) (since \( 1 + \frac{2i}{n} \) ranges from 1 to 3 as \( i \) goes from 1 to \( n \)).

### Step 2: Write the Integral
The Riemann sum approximates the following integral as \( n \to \infty \):

\[
\int_{1}^{3} \left( x \right)^{10} \, dx
\]

### Step 3: Solve the Integral
Now, compute the integral:

\[
\int_{1}^{3} x^{10} \, dx = \left[ \frac{x^{11}}{11} \right]_{1}^{3}
\]

Evaluating the integral:

\[
= \frac{3^{11}}{11} - \frac{1^{11}}{11}
= \frac{3^{11} - 1}{11}
\]

### Final Answer:
Thus, the value of the original limit is:

\[
\boxed{\frac{3^{11} - 1}{11}}
\]

Would you like a detailed explanation of any step, or do you have any questions?

Here are five related questions to deepen your understanding:
1. How do Riemann sums relate to definite integrals?
2. What are the general steps to converting a sum to an integral?
3. How does the behavior of functions change as \( n \to \infty \)?
4. Can all sums involving limits be converted into integrals?
5. How does the power of 10 in \( (1 + \frac{2i}{n})^{10} \) affect the limit?

**Tip:** Converting sums to integrals is a powerful method in calculus, particularly when dealing with limits that grow large!

lim⁡𝑛→∞ ∑𝑖=1𝑛 1/𝑛 (1 + 2𝑖/𝑛)^{10}

Explore the solution to the limit problem involving Riemann sums, showing how to convert the sum into a definite integral as n approaches infinity. This calculus problem illustrates key steps, including the use of Riemann sums, integration, and limits.

Math Problem Statement

lim ⁡ 𝑛 → ∞ ∑ 𝑖

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