How to merge the sum and ##x^n##?

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In summary, the conversation discusses merging the terms ##x^n## and ##\displaystyle\sum^x_{k=1} \frac{d^k}{d^kx} \frac{x^ny^k}{k}## by changing the denominator of the summand and expanding the summation. It is mentioned that the upper bound of the summation should not be x, and the derivative notation should be corrected. The final result is determined to be equal to ##(x+y)^n##.
  • #1
MevsEinstein
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What the title says
How do I merge ##x^n + \displaystyle\sum^x_{k=1} \frac{d^k}{d^kx} \frac{x^ny^k}{k}##? I tried changing the denominator of the summand to ##k+1## and make ##k## go from zero, but I had to divide by zero when k equaled one.
 
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  • #2
MevsEinstein said:
Summary: What the title says

How do I merge ##x^n + \displaystyle\sum^x_{k=1} \frac{d^k}{d^kx} \frac{x^ny^k}{k}##? I tried changing the denominator of the summand to ##k+1## and make ##k## go from zero, but I had to divide by zero when k equaled one.
Is the upper bound index of your sum right? With to merge both terms, you mean to write everything as a unique sum?
 
  • #3
I'm not able to guess what you mean by this expression -- ##x^n + \displaystyle\sum^x_{k=1} \frac{d^k}{d^kx} \frac{x^ny^k}{k}##. The upper limit of the summation should not be x, and it should be something other than n.

Also, and this is minor, your derivative is not formed correctly. The usual notation for the k-th derivative operator with respect to x is ##\frac {d^k}{dx^k}##.

Once you get the summation written correctly, the ##x^n## term outside the summation would get added to the corresponding term(s) of the summation. You'll probably need to expand the summation to get the addition right. Sums like this show up in power series solutions of differential equations.
 
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  • #4
Assuming your sum goes from 1 to n: While it's possible to write this as ##\displaystyle \sum_{k=0}^n ...## it will look ugly. What's the point of this transformation?
 
  • #5
Hold up the upper bound is ##n##. I couldn't edit though.
 
  • #6
LCSphysicist said:
you mean to write everything as a unique sum?
I wanted to get rid of ##x^n## and put something in replacement inside the summand of the sum so that the expression is the same. And my upper bound was supposed to be ##n##, sorry for the misconvenience. I couldn't edit for some reason.
 
  • #7
Ah, I missed that the yk term disappears on its own for k=0 in my previous post. Then it's only mildly ugly.
$$x^n + \sum^n_{k=1} \frac{d^k}{dx^k} \frac{x^ny^k}{k} = \sum^n_{k=0} \frac{d^k}{dx^k} \frac{x^ny^k}{|k-1/2|+1/2}$$

The denominator can be replaced by max(1,k).
 
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  • #8
mfb said:
$$x^n + \sum^n_{k=1} \frac{d^k}{d^kx} \frac{x^ny^k}{k} = \sum^n_{k=0} \frac{d^k}{d^kx} \frac{x^ny^k}{|k-1/2|+1/2}$$
Minor nit: the derivatives in the above should be
mfb said:
$$x^n + \sum^n_{k=1} \frac{d^k}{dx^k} \frac{x^ny^k}{k} = \sum^n_{k=0} \frac{d^k}{dx^k} \frac{x^ny^k}{|k-1/2|+1/2}$$
 
  • #9
But
$$ x^n + \sum^n_{k=1} \frac{d^k}{dx^k} \frac{x^ny^k}{k} $$
is simpler than
$$ \sum^n_{k=0} \frac{d^k}{dx^k} \frac{x^ny^k}{|k-1/2|+1/2} $$
so why not stick with that?

Anyway, what is ## y ## ?
 
  • #10
pbuk said:
But
$$ x^n + \sum^n_{k=1} \frac{d^k}{dx^k} \frac{x^ny^k}{k} $$
is simpler than
$$ \sum^n_{k=0} \frac{d^k}{dx^k} \frac{x^ny^k}{|k-1/2|+1/2} $$
so why not stick with that?

Anyway, what is ## y ## ?
Actually let's just stick with that. The expression is equal to \(\displaystyle (x+y)^n\). Turns out that this formula is harder to use than the one we have now.
 
  • #11
Mark44 said:
Minor nit: the derivatives in the above should be
Copy&paste error, fixed.

@pbuk: That's my point, the original expression is easier.
MevsEinstein said:
The expression is equal to \(\displaystyle (x+y)^n\).
Only if you make it a k! in the denominator.
 
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1. How do you merge the sum and ##x^n##?

To merge the sum and ##x^n##, you can use the summation notation, which is represented by the Greek letter sigma (Σ). This notation allows you to write out the terms of a sum and then use the upper and lower limits to indicate the range of values to be summed. For example, the sum of ##x^n## from 1 to 5 can be written as Σn=15 x^n.

2. What is the purpose of merging the sum and ##x^n##?

Merging the sum and ##x^n## allows you to simplify and manipulate mathematical expressions involving summation. It also helps in finding patterns and making predictions about the behavior of a sequence or series.

3. Can the order of terms be changed when merging the sum and ##x^n##?

Yes, the order of terms can be changed when merging the sum and ##x^n##. The commutative property of addition allows you to rearrange the order of terms without changing the result of the sum.

4. Are there any rules or formulas for merging the sum and ##x^n##?

Yes, there are several rules and formulas that can be used when merging the sum and ##x^n##. Some of these include the distributive property, the associative property, and the power rule for exponents.

5. How can merging the sum and ##x^n## be applied in real-life situations?

Merging the sum and ##x^n## can be applied in various fields such as physics, economics, and engineering. For example, it can be used to calculate the total cost of a project with changing costs over time, or to determine the total distance traveled by an object with changing speed over time.

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