Finding Taylor Series of Functions - Tips to Make it Easier

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To find the Taylor series of functions like f(x) = ln(x) about x = e, it's crucial to ensure the function is analytic at that point. Start by calculating the function's value and its derivatives at x = e, such as f(e), f'(e), and f''(e). Look for patterns in the derivatives to help construct the series. Differentiating power functions is straightforward, making these problems more manageable. Following these steps can simplify the process of finding Taylor series.
pnazari
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I was wondering if someone can give me some tips for finding the taylor series of functions. For example this was a test question we had:

Find the taylor series of f(x)=ln(x) about x=e

I know how to start it off but I get confused halfway through and can't seem to figure out what to do. Are there some simple steps? Any help/tips will be appreciated.

For me, this is the hardest section of Calc. II
 
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There aren't really any advice here.You just make sure the function is analytical in that point,which,in this case is...

So differentiate and write that series...

Daniel.
 
Find as much information as you can about f(x), usually f(e), f'(e) f''(e) if u can get that far. then find a pattern from there and write a series.
 
Thankfully,the power function can easily differentiated any # of times,so this problem is really simple.


Daniel.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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