Math Problem Statement

explain this proof please

explain this proof please

Solution

This proof demonstrates that 1+x<ex1 + x < e^x by utilizing the definition of exe^x as a limit and establishing an increasing sequence that approaches exe^x. Let's break down each part:

  1. Starting Point and Objective:
    The goal is to show that 1+x<ex1 + x < e^x.

  2. Definition of exe^x:
    The proof uses the limit definition of exe^x: ex=limn(1+xn)ne^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n This expression represents exe^x as the limit of a sequence.

  3. Intermediate Step:
    The proof constructs an inequality to compare (1+xn)n\left(1 + \frac{x}{n}\right)^n to other terms that help build the increasing sequence. It claims: (1+xn)n<(1+2x2n+x24n2)n=(1+x2n)2n\left(1 + \frac{x}{n}\right)^n < \left(1 + 2\frac{x}{2n} + \frac{x^2}{4n^2}\right)^n = \left(1 + \frac{x}{2n}\right)^{2n} This inequality builds a comparison between terms by introducing approximations to simplify the sequence while preserving the limit.

  4. Increasing Sequence:
    The expression (1+x2n)2n\left(1 + \frac{x}{2n}\right)^{2n} becomes part of a sequence. We observe that: 1+x<(1+x2)2<ex1 + x < \left(1 + \frac{x}{2}\right)^2 < \cdots \to e^x This chain of inequalities shows that 1+x1 + x is smaller than terms in an increasing sequence that eventually converges to exe^x.

  5. Conclusion:
    Since 1+x1 + x is the beginning of an increasing sequence that tends to exe^x, we conclude that 1+x<ex1 + x < e^x.

This proof effectively uses the limit definition of exe^x and builds a sequence of inequalities to show that 1+x1 + x is always less than exe^x because it lies at the beginning of a sequence that approaches exe^x.

Would you like further details on any specific part of this proof?

Here are five related questions to expand understanding:

  1. How is the limit definition of exe^x derived?
  2. Why does the sequence (1+x2n)2n\left(1 + \frac{x}{2n}\right)^{2n} converge to exe^x?
  3. What other inequalities can we use to bound exe^x from below?
  4. Why does proving 1+x1 + x as part of an increasing sequence lead to the conclusion 1+x<ex1 + x < e^x?
  5. Can this technique be generalized to prove inequalities for other exponential functions?

Tip: When working with limits and inequalities, constructing sequences that approach a target value can help establish bounds for expressions.

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Math Problem Analysis

Mathematical Concepts

Calculus
Exponential Function
Limits
Inequalities

Formulas

e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n

Theorems

Limit Definition of Exponential Function

Suitable Grade Level

College Level