Indeterminate Forms: Converging to e & 1

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In summary, indeterminate forms are mathematical expressions that cannot be evaluated directly and typically arise when dividing by zero or taking the limit of a function that approaches infinity or negative infinity. Convergence in mathematics means that a sequence or function approaches a specific value as its input increases or decreases, and in the case of indeterminate forms converging to e and 1, this means that the limit of the function approaches these values as the input approaches infinity or negative infinity. The numbers e and 1 are commonly involved in indeterminate forms due to their special role in mathematics, and to evaluate indeterminate forms, one must use mathematical techniques such as L'Hôpital's rule, substitution, or algebraic manipulation. Indeterminate forms have many applications in
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I know [tex] \lim_{n \rightarrow \infty} (1 + 1/n)^n = \lim_{n \rightarrow \infty} 1^{\infty} [/tex], which is an indeterminate form, converging to e in this case. But what if the original sequence is [tex] a_n = 1^n [/tex]. Then as n tends to infinity, the function converges to 1 (because it's constant and the limit of a constant function is any term of the sequence). Is my reasoning correct here?EDIT: The original sequence is (1 + 1/n)^n, I messed up my latex.
 
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Yes, you are correct.
 

Related to Indeterminate Forms: Converging to e & 1

What are indeterminate forms?

Indeterminate forms are mathematical expressions that cannot be evaluated directly. They typically arise when dividing by zero or taking the limit of a function that approaches infinity or negative infinity.

What does it mean to converge to a specific value?

Convergence in mathematics means that a sequence or function approaches a specific value as its input increases or decreases. In the case of indeterminate forms converging to e and 1, it means that the limit of the function approaches these values as the input approaches infinity or negative infinity.

Why do indeterminate forms often involve the numbers e and 1?

The numbers e and 1 are commonly involved in indeterminate forms because they are special values that are often used in mathematics. For example, e is the base of the natural logarithm and often appears in exponential functions, while 1 is the identity element for multiplication and division.

How do you evaluate indeterminate forms?

To evaluate indeterminate forms, you need to use mathematical techniques such as L'Hôpital's rule, substitution, or algebraic manipulation. These methods allow you to simplify the expression and determine the limit of the function as the input approaches a specific value, such as e or 1.

What are the applications of indeterminate forms in science?

Indeterminate forms have many applications in science, particularly in calculus and physics. They are used to model real-life situations that involve rates of change or limits, such as growth and decay, motion, and optimization. Indeterminate forms also play a crucial role in understanding and solving differential equations, which are used to describe various phenomena in science and engineering.

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