- #1
JonF
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Does
[tex]\lim_{n \rightarrow \infty} \sum^{n}_{i=1} 1/n [/tex]
Equal: one, zero, or something else?
[tex]\lim_{n \rightarrow \infty} \sum^{n}_{i=1} 1/n [/tex]
Equal: one, zero, or something else?
arildno said:Really?
[tex]\sum_{i=1}^{n}\frac{1}{n}=1[/tex]
IMHO..
JonF said:then does
[tex]\lim_{n \rightarrow \infty} \sum^{n}_{i=1} \frac{2}{n} = 2[/tex]
matt grime said:however the error is thinking that you may let n go to infinity in the denominator independently of the limit of the sum.
master_coda said:This is the heart of the problem. When you take the limit of something, you have to consider that something as a whole. People sometimes make a similar mistake when they consider . You can't take the limit of the value inside the brackets first and then apply the exponent and conclude that the result is 1. Similarly you can't just take the limit of a value inside a summation and then apply the summation. Sometimes that will give you the correct answer, but that does not work in general.
JonF said:Thank you. That makes a lot of sense. But then would?
[tex]\lim_{n \rightarrow \infty}\sum^{n}_{i=1} \frac{1}{n-1} = 1[/tex]
Gokul43201 I am not sure what you are asking…
JonF said:Thank you. That makes a lot of sense. But then would?
[tex]\lim_{n \rightarrow \infty}\sum^{n}_{i=1} \frac{1}{n-1} = 1[/tex]
[tex]\lim_{n\rightarrow\infty} \sum_{i=1}^{2n} \frac{1}{n}=2[/tex]JonF said:Ok, how about this one. Since [tex]\sum_{i=1}^{2n}\frac{1}{n}=2[/tex] what does [tex]\lim_{n\rightarrow\infty} \sum_{i=1}^{2n} \frac{1}{n}=[/tex]
Then don't worry about itJonF said:No typo.
These questions stem from a thread that was around a several weeks ago, where it was stated that [tex]\frac{1}{\infty}=0[/tex]. My real question that I’ve been building up to is: how can [tex]\lim_{n \rightarrow \infty}\sum^{n}_{i=1} 0 = 1[/tex]
This thread is 6 years old, Halls!HallsofIvy said:Then don't worry about it
[tex]\frac{1}{\infty}= 0[/tex]
is not true in standard analysis.
Infinitely small numbers, also known as infinitesimals, are numbers that are smaller than any positive real number but are still greater than zero. They are used in calculus and other branches of mathematics to represent quantities that are approaching zero, but never actually reach it.
Infinitely small numbers are different from zero because they have a non-zero value, even though it is incredibly small. Zero is a well-defined number that represents the absence of a quantity, while infinitesimals represent quantities that are approaching zero but never actually reach it.
Yes, infinitely small numbers are real numbers. They are part of the real number system and can be represented on a number line just like any other real number. However, they are not considered to be "normal" real numbers because they are smaller than any positive real number.
Yes, infinitesimals can be used in calculations, particularly in calculus. They are used to represent quantities that are approaching zero, and can be manipulated using the same rules as other real numbers. However, they cannot be used in all mathematical operations and must be handled carefully to avoid mathematical errors.
Infinitely small numbers are important in mathematics because they allow us to describe and analyze quantities that are approaching zero, but never actually reach it. This is particularly useful in calculus, where infinitesimals are used to define derivatives and integrals. They also have applications in other areas of mathematics, such as in the study of limits and continuity.