whatisreality said:Homework Statement
I've been given the series 3,4,6,10,18... and asked to express as the ∑ar, between r=0 and r = infinity.
Homework Equations
The Attempt at a Solution
Well, I can see a pattern! The difference between terms doubles every time. I'm having difficulty expressing this mathematically though... I think it should be 3 plus some function of r. Is there a technique for this? All we've been told is to do it 'by inspection'.
That's doing it the hard way! Reminds me of that story about John Nash solving the back-and-forth fly problem by summing the series...Ray Vickson said:The differences double, so the terms are 3, 3+1, 3+1+2, 3+1+2+4, 3+1+2+4+8,...
haruspex said:That's doing it the hard way! Reminds me of that story about John Nash solving the back-and-forth fly problem by summing the series...
I'm confused. I want an expression for ar using sigma notation. I thought that's what my question asked. What does my question actually mean then? As in, I thought I stated the problem correctly, AND the problem was I want an expression using sigma notation.Ray Vickson said:Might not be the hard way if the question was not really stated correctly by the OP; have we not seen that many times before in this forum? If the OP gave the correct statement, then, of course, you are right; but if the question really wanted an expression for ##a_r## in sigma notation, then what I outlined would go part way to the solution.
whatisreality said:I'm confused. I want an expression for ar using sigma notation. I thought that's what my question asked. What does my question actually mean then? As in, I thought I stated the problem correctly, AND the problem was I want an expression using sigma notation.
whatisreality said:I've been given the series 3,4,6,10,18... and asked to express as the ∑ar, between r=0 and r = infinity.
Yes, it's not very clear. Because it referred to the given numbers as a series (which to me implies summation), not a sequence, I was looking for a general closed form for the numbers ##a_0 = 3, a_1=4, ##... so that the series sum can be written ##\Sigma a_r##.Ray Vickson said:Might not be the hard way if the question was not really stated correctly by the OP; have we not seen that many times before in this forum? If the OP gave the correct statement, then, of course, you are right; but if the question really wanted an expression for ##a_r## in sigma notation, then what I outlined would go part way to the solution.
Sigma notation is a mathematical shorthand used to express a series in a more compact form. It is represented by the Greek letter sigma (Σ) and allows us to write a sum of terms in a concise way.
To write a series in sigma notation, we use the following format: Σi=kn ai, where i is the index variable, k is the starting value, n is the ending value, and ai represents the terms in the series.
The main purpose of using sigma notation is to simplify and condense the notation of a series, making it easier to read and work with. It also allows us to express infinite series and series with a large number of terms in a more manageable way.
To evaluate a series given in sigma notation, we substitute the values of the index variable i into the expression ai and sum up all the resulting terms. This will give us the final value of the series.
Yes, a series can be written in multiple ways using sigma notation. For example, we can change the starting and ending values, the index variable, or the expression for the terms. However, all these notations will represent the same series and will have the same final value when evaluated.