Why isn't this zero? Find the limit 1^2/n^3+2^2/n^3+...+n^2/n^3. Reddit calculus r/HomeworkHelp
TLDRThe video script discusses the common misconception of calculating limits as n approaches infinity for a series of expressions. The instructor emphasizes the importance of not simply adding zeros when dealing with infinity and introduces two methods for finding the correct limit: algebraic manipulation and calculus through integration. The script concludes with the correct approach using an algebraic formula to find the limit, resulting in the answer of 2/3. The concept is further illustrated with an integral representing the area under a curve, reinforcing the understanding of limits in calculus.
Takeaways
- 🚫 Avoid the common mistake of naively substituting infinity for very large values in limits.
- ✅ Always consider the number of terms in a series when dealing with limits involving infinity.
- 🔢 Factor out common terms, like n^3 from the denominator, to simplify the limit expression.
- 📈 Utilize known formulas for sums of series, such as the sum of the first n squares, to find limits.
- 📚 Apply algebraic techniques before taking the limit as n approaches infinity to simplify the problem.
- 🌟 Recognize that the limit can represent physical concepts, like areas under curves in calculus.
- 📈 Convert series into integrals when appropriate to compute limits using calculus methods.
- 🔍 Look for the highest power of n in both the numerator and denominator when evaluating limits of rational functions.
- 👌 Simplify the final expression by canceling out terms and reducing fractions.
- 🎓 Understanding limits is crucial for a solid foundation in calculus and its applications.
- 🔗 The solution to the given limit problem is related to the area under the curve of x^2 from 0 to 1, which is 1/3.
Q & A
What is the common mistake discussed at the beginning of the transcript?
-The common mistake discussed is incorrectly assuming that the limit of a long expression with infinite terms, each having a factor of infinity or 1/infinity, simplifies to zero. The speaker emphasizes the importance of careful analysis instead of jumping to conclusions.
Why is it incorrect to simply add up zeros from each term when taking the limit as n approaches infinity?
-Adding up zeros from each term would be incorrect because when dealing with limits, the expression does not simply become a sum of zeros. The presence of infinity in the terms means that the limit results in an indeterminate form, which cannot be concluded without proper analysis.
What is the first method mentioned for solving the limit as n approaches infinity?
-The first method mentioned is to first do the algebraic manipulation by factoring out common terms and then taking the limit as n approaches infinity.
What is the second method for solving the limit, and how is it related to calculus?
-The second method is recognizing the limit of a series as a Riemann sum and converting it into an integral. This method is related to calculus as it involves the computation of integrals to evaluate the limit.
What is the formula for the sum of the first n squares?
-The formula for the sum of the first n squares is given by (n(n + 1)(2n + 1))/6. This formula allows for the calculation of the sum of squares from 1^2 to n^2.
How does the speaker verify the formula for the sum of the first n squares?
-The speaker verifies the formula by plugging in the value of n as 3 and manually calculating the sum of squares from 1 to 3, which matches the result obtained from the formula.
What is the final result of the limit as n approaches infinity for the given expression?
-The final result of the limit as n approaches infinity for the given expression is 2/6, which simplifies to 1/3.
How does the limit relate to the area under a curve in the context of calculus?
-The limit represents the area under the curve y = x^2 from 0 to 1. By evaluating the definite integral of x^2 from 0 to 1, the result is the same as the limit, which is 1/3.
What is the antiderivative of x^2 used to evaluate the area under the curve?
-The antiderivative of x^2 is (1/3)x^3. This is used to evaluate the definite integral to find the area under the curve from 0 to 1.
Why is it important to reduce fractions in calculus, as mentioned in the transcript?
-Reducing fractions in calculus is important for simplifying expressions and obtaining the most accurate and clear results. It helps in presenting the final answer in its simplest form, which is easier to understand and interpret.
How does the speaker suggest approaching limits involving rational functions when n approaches infinity?
-The speaker suggests that for limits involving rational functions when n approaches infinity, one can either pick up the highest power of n from both the numerator and the denominator to quickly find the limit or divide the entire expression by the highest power of n to simplify it before evaluating the limit.
Outlines
📚 Understanding Common Mistakes in Calculating Limits
The paragraph discusses a common misconception in calculating limits as n approaches infinity. It emphasizes the mistake of naively substituting infinity for the variable in a series, which can lead to incorrect conclusions. The speaker clarifies that simply adding zeros does not yield a meaningful limit and highlights the importance of considering the number of terms in the series. The correct approach involves factoring out the common denominator (n^3) and applying a known formula for the sum of the first n squares to find the limit. The explanation concludes with a reminder to be cautious when dealing with indeterminate forms and the importance of algebraic manipulation in finding limits.
📈 Applying a Formula for the Sum of Squares
This paragraph delves into the application of a mathematical formula for the sum of the first n squares, which is crucial for finding the limit as n approaches infinity. The speaker demonstrates how to use the formula by plugging in values and verifying its correctness with a specific example. The process involves simplifying the expression by canceling terms and focusing on the dominant terms as n becomes very large. The final answer is obtained by reducing the fraction and is related to the area under a curve in calculus, specifically the area between x=0 and x=1 for the function y=x^2. This connection between limits and integral calculus is briefly mentioned, with an encouragement to explore further if the topic is familiar.
Mindmap
Keywords
💡limit
💡infinity
💡algebra
💡calculus
💡Riemann sum
💡indeterminant form
💡factor out
💡sum of squares
💡formula
💡rational function
💡area under a curve
Highlights
The importance of being cautious when evaluating limits as n approaches infinity, to avoid common misconceptions.
The misconception of simply plugging in infinity and getting zero for all terms in the limit expression.
The correct approach involves factoring out common terms and simplifying the expression before taking the limit.
The sum of the first n squares has a known formula, which is a key insight for solving the problem algebraically.
The formula for the sum of the first n squares is (n(n + 1)(2n + 1))/6, which is a crucial result for the problem.
Verification of the formula using a specific value of n (e.g., 3) to demonstrate its validity.
The process of simplifying the limit expression by canceling terms and focusing on the dominant terms as n approaches infinity.
The final answer for the limit, which is 2/6 or 1/3, showing the relationship between the sum of squares and the limit.
The connection between the limit problem and the area under a curve, specifically y = x^2 from 0 to 1.
The method of evaluating the integral using antiderivatives and applying the fundamental theorem of calculus.
The integral method as an alternative way to solve the limit problem, offering a different perspective on the problem.
The importance of reducing fractions in calculus for obtaining the simplest form of the solution.
The practical application of the limit in calculating the area under a curve, demonstrating the usefulness of the concept.
The detailed explanation of the steps involved in evaluating the integral, including finding the antiderivative and applying limits.
The final result of the integral, which matches the limit result, confirming the correctness of the algebraic and calculus methods.
The emphasis on the importance of understanding both algebraic and calculus methods for solving limit problems.
Transcripts
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