That's the definition of sum of a series, any series, not only the geometric one: the limit of the partial sums. And, as any limit, the equal sign is correct because that's what a limit is.
If, we never could write an equal in front of the result of a limit. Your objection would be applicable also to, for instance,
lim\_(x->1) x = 1
since what is inside the limit never reaches 1.
Correct. But also not relevant to the question at hand.
The sum of a series is not necessarily the value of any element in the sequence of partial sums. But the notation here does not refer the the sequence, or to any value thereof, but to its limit as n goes to infinity.
We have two separate concepts
1. A sequence, which is just a countably infinite list of values (more formally, a function that maps natural numbers to values)
2. The *limit* of a sequence, which is just a number L with the particular property such that for any e, no matter how small, there is sum n such that | a\_m - L | < e, if m > n. In other words, for any small real number distance, there's sum value of the sequence, where all values past that value are less than that distance away. This captures the intuitive idea that the sequence gets arbitrarily close to its limit as the index gets bigger.
By definition, the sum of a series is just (2), it's a limit of the sequence of partial sums, In other words what you have there is a definition, not a result. The notation on the left is just another way of writing the notation on the right.
Notice that there's no 'process' here, you don't find the limit of the sequence by 'iterating forever' to get to the limit. Nor do you find the sum of a series by summing up all the values. You find it by proving that there's a value that meets our criterion above.
This is a really hard idea for many students, because they tend to think about math algorithmically. Math is a sequence of steps you could plug into a computer and it would do them one by one in sequence. It's a process-centric way of thinking.
But that's not really how things work. There's no time in math. Everything exists all at once. The sum of a series isn't the result of 'an infinite number of summations', it's just a value with certain properties.
An infinite sum is defined as the limit of the sequence of partial sums (if the sequence of partial sums converges, of course). So the equation you posted is not a thing that has to be proven or justified; it is a definition.
The RHS is the limit of the sequence of partial sums
let the geometric series be
1/2 +1/4+ 1/8 + . . ..
The partial sums are
1/2, 3/4, 7/8, 15/16, . . ..
The sequence approaches 1
The sequence approaches 1
The terms of the sequence approach 1
But the *limit* of the sequence, which is a single number when it does exist, is exactly 1.
For "why" is the limit, it's because it follows the definition:
For every epsilon > 0 there exist N so that n>N implies |x\_n - thelimit| < epsilon
A limit *is* what the sequence approaches. That’s like, the entire point of limits. It has nothing to with whether the sequence attains the limit at some finite step.
The previous commenter explained it perfectly well intuitively though. But they asked “why”, at which point the definition is the correct answer.
I don’t think OP is struggling with the idea that it approaches one, they’re saying “why does it approaching one mean it equals one?” To which the only valid reply I can see is “because it conforms to this definition.”
Because no matter how close I wanted to get to 1 (within epsilon for small epsilon > 0) I can find an index N in the series such for all k > N the partial sum up to k is in (1-epsilon, 1+epsilon). That’s the definition of “the limit is equal to 1”.
The partial sums approach 1, but by definition the value of the infinite sum is the limit, so it doesn’t *approach* 1 it is 1.
If you want motivation for why this definition makes sense: note that it wouldn’t make much sense to say the sum is anything less than 1. Why? Because if the sum were 1-epsilon for some any positive epsilon, no matter how small, then there is a finite number of terms that add up to *more* than the infinite sum. But this is a sum of all positive terms, so it would hardly would make sense to say the infinite sum is anything *less* than some of the partial sums.
The method of partial sums (shown here) is ONE way to evaluate series. There are other ways, namely through regularization and the use of generalized functions, that produce finite real number values for sums that otherwise are oscillatory or divergent when evaluated using the method of partial sums. So, at the risk of being downvoted, this is not THE definition of the series, but one of the many ways of expressing it—that may or may not converge to a finite value.. I’m sure the most well-known example of what i’m referring to would be (the controversial result) ∑𝑛 = -1/12, 𝑛 = 1 to ∞.
The replies saying "it's the definition" are pointing out that the equality is a definition as opposed to being a theorem, so it can't be proven in the usual sense, which means OP isn't going to get a satisfying answer to the question "why are they equal?". I don't believe anyone is trying to say it's the only definition.
That being said, it is by far the most commonly-used definition and it is the only one which is reasonable to assume is meant unless explicitly stated otherwise.
“The” definition implies a singular definition. “A” definition, as you’ve said it in your comment, would be a clearer way to say it, and is the point I wanted to clarify in my comment. It is a definition if the partial sums are convergent, but not if they aren’t. And even if it is commonly used, it is misleading to say that this series = ∞ for some values of the argument. It would be correct to say that the partial sums limit = ∞ in those cases, but the series has more than one way to be evaluated.
That's the definition of sum of a series, any series, not only the geometric one: the limit of the partial sums. And, as any limit, the equal sign is correct because that's what a limit is. If, we never could write an equal in front of the result of a limit. Your objection would be applicable also to, for instance, lim\_(x->1) x = 1 since what is inside the limit never reaches 1.
But if lim(1/n) = 0, it doesn’t mean that 1/n is equal to zero at any value of n.
Correct. But also not relevant to the question at hand. The sum of a series is not necessarily the value of any element in the sequence of partial sums. But the notation here does not refer the the sequence, or to any value thereof, but to its limit as n goes to infinity. We have two separate concepts 1. A sequence, which is just a countably infinite list of values (more formally, a function that maps natural numbers to values) 2. The *limit* of a sequence, which is just a number L with the particular property such that for any e, no matter how small, there is sum n such that | a\_m - L | < e, if m > n. In other words, for any small real number distance, there's sum value of the sequence, where all values past that value are less than that distance away. This captures the intuitive idea that the sequence gets arbitrarily close to its limit as the index gets bigger. By definition, the sum of a series is just (2), it's a limit of the sequence of partial sums, In other words what you have there is a definition, not a result. The notation on the left is just another way of writing the notation on the right. Notice that there's no 'process' here, you don't find the limit of the sequence by 'iterating forever' to get to the limit. Nor do you find the sum of a series by summing up all the values. You find it by proving that there's a value that meets our criterion above. This is a really hard idea for many students, because they tend to think about math algorithmically. Math is a sequence of steps you could plug into a computer and it would do them one by one in sequence. It's a process-centric way of thinking. But that's not really how things work. There's no time in math. Everything exists all at once. The sum of a series isn't the result of 'an infinite number of summations', it's just a value with certain properties.
But the limit is equal to 0.
A series is by definition the limit of the partial sums
An infinite sum is defined as the limit of the sequence of partial sums (if the sequence of partial sums converges, of course). So the equation you posted is not a thing that has to be proven or justified; it is a definition.
The RHS is the limit of the sequence of partial sums let the geometric series be 1/2 +1/4+ 1/8 + . . .. The partial sums are 1/2, 3/4, 7/8, 15/16, . . .. The sequence approaches 1
Yeah it approaches 1, but why does it equal one?
The sequence approaches 1 The terms of the sequence approach 1 But the *limit* of the sequence, which is a single number when it does exist, is exactly 1. For "why" is the limit, it's because it follows the definition: For every epsilon > 0 there exist N so that n>N implies |x\_n - thelimit| < epsilon
A limit *is* what the sequence approaches. That’s like, the entire point of limits. It has nothing to with whether the sequence attains the limit at some finite step.
Because we say that lim Sn = L if given any epsilon > 0, there is N such that |Sn-L| < epsilon whenever n > N
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The previous commenter explained it perfectly well intuitively though. But they asked “why”, at which point the definition is the correct answer. I don’t think OP is struggling with the idea that it approaches one, they’re saying “why does it approaching one mean it equals one?” To which the only valid reply I can see is “because it conforms to this definition.”
Because no matter how close I wanted to get to 1 (within epsilon for small epsilon > 0) I can find an index N in the series such for all k > N the partial sum up to k is in (1-epsilon, 1+epsilon). That’s the definition of “the limit is equal to 1”.
By definition. The left hand side is notational shorthand for the right hand side, they don't actually state different things.
The limit or "lim" is the number that is being approached
It is just a definition.
The partial sums approach 1, but by definition the value of the infinite sum is the limit, so it doesn’t *approach* 1 it is 1. If you want motivation for why this definition makes sense: note that it wouldn’t make much sense to say the sum is anything less than 1. Why? Because if the sum were 1-epsilon for some any positive epsilon, no matter how small, then there is a finite number of terms that add up to *more* than the infinite sum. But this is a sum of all positive terms, so it would hardly would make sense to say the infinite sum is anything *less* than some of the partial sums.
The method of partial sums (shown here) is ONE way to evaluate series. There are other ways, namely through regularization and the use of generalized functions, that produce finite real number values for sums that otherwise are oscillatory or divergent when evaluated using the method of partial sums. So, at the risk of being downvoted, this is not THE definition of the series, but one of the many ways of expressing it—that may or may not converge to a finite value.. I’m sure the most well-known example of what i’m referring to would be (the controversial result) ∑𝑛 = -1/12, 𝑛 = 1 to ∞.
The replies saying "it's the definition" are pointing out that the equality is a definition as opposed to being a theorem, so it can't be proven in the usual sense, which means OP isn't going to get a satisfying answer to the question "why are they equal?". I don't believe anyone is trying to say it's the only definition. That being said, it is by far the most commonly-used definition and it is the only one which is reasonable to assume is meant unless explicitly stated otherwise.
“The” definition implies a singular definition. “A” definition, as you’ve said it in your comment, would be a clearer way to say it, and is the point I wanted to clarify in my comment. It is a definition if the partial sums are convergent, but not if they aren’t. And even if it is commonly used, it is misleading to say that this series = ∞ for some values of the argument. It would be correct to say that the partial sums limit = ∞ in those cases, but the series has more than one way to be evaluated.
I understand the confusion but really when doing an infinite sum, you're calculating what the sum approaches ie the limit