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Checking my understanding of the process of developing function into power series

0

0

$begingroup$

I would like to get some help with the next problem:

I'm trying to learn how to develop the real function into the power series. After reading my book, i want to check if i understood correctly what is the process of the developing of the given real function. So, this is how i understand what i need to do in order to develop the given real function f:

1) Check if the given function f is infinitely diferentiable and where.

2) Chose the point $x_0$ in which we are going to develop the function.

3) Check if the given function f is continuous with all of its derivatives, up to the $n$-th order, in some neighborhood of the point $x_0$. If this is fulfilled, we have that we can write
$$f(x) = P_n(x, x_0) + R_n(x).$$

4) Check if $limlimits_n to infty R_n(x) = 0$.

5) Check if $f(x_0) = P_n(x, x_0) = sum_n = 0^infty fracf^(n)(x_0)n!(x - x_0^n)$. This means that we have to check the convergence of the Taylor series that we got and to calculate the sum of the series if the series is convergent.

6) If all conditions are fulfilled, than we can say that function $f$ can be developed into power series $sum_n = 0^infty fracf^(n)(x_0)n!(x - x_0^n)$ and we can call that function analytic.

Please, could you tell me if i understand this process correctly and if not where do i make a mistake?

real-analysis power-series

share|cite|improve this question

asked Mar 25 at 19:25

SlowLearnerSlowLearner

18513

$endgroup$

add a comment | 

0

0

$begingroup$

I would like to get some help with the next problem:

I'm trying to learn how to develop the real function into the power series. After reading my book, i want to check if i understood correctly what is the process of the developing of the given real function. So, this is how i understand what i need to do in order to develop the given real function f:

1) Check if the given function f is infinitely diferentiable and where.

2) Chose the point $x_0$ in which we are going to develop the function.

3) Check if the given function f is continuous with all of its derivatives, up to the $n$-th order, in some neighborhood of the point $x_0$. If this is fulfilled, we have that we can write
$$f(x) = P_n(x, x_0) + R_n(x).$$

4) Check if $limlimits_n to infty R_n(x) = 0$.

5) Check if $f(x_0) = P_n(x, x_0) = sum_n = 0^infty fracf^(n)(x_0)n!(x - x_0^n)$. This means that we have to check the convergence of the Taylor series that we got and to calculate the sum of the series if the series is convergent.

6) If all conditions are fulfilled, than we can say that function $f$ can be developed into power series $sum_n = 0^infty fracf^(n)(x_0)n!(x - x_0^n)$ and we can call that function analytic.

Please, could you tell me if i understand this process correctly and if not where do i make a mistake?

real-analysis power-series

share|cite|improve this question

asked Mar 25 at 19:25

SlowLearnerSlowLearner

18513

$endgroup$

add a comment | 

$begingroup$

I would like to get some help with the next problem:

I'm trying to learn how to develop the real function into the power series. After reading my book, i want to check if i understood correctly what is the process of the developing of the given real function. So, this is how i understand what i need to do in order to develop the given real function f:

1) Check if the given function f is infinitely diferentiable and where.

2) Chose the point $x_0$ in which we are going to develop the function.

3) Check if the given function f is continuous with all of its derivatives, up to the $n$-th order, in some neighborhood of the point $x_0$. If this is fulfilled, we have that we can write
$$f(x) = P_n(x, x_0) + R_n(x).$$

4) Check if $limlimits_n to infty R_n(x) = 0$.

5) Check if $f(x_0) = P_n(x, x_0) = sum_n = 0^infty fracf^(n)(x_0)n!(x - x_0^n)$. This means that we have to check the convergence of the Taylor series that we got and to calculate the sum of the series if the series is convergent.

6) If all conditions are fulfilled, than we can say that function $f$ can be developed into power series $sum_n = 0^infty fracf^(n)(x_0)n!(x - x_0^n)$ and we can call that function analytic.

Please, could you tell me if i understand this process correctly and if not where do i make a mistake?

real-analysis power-series

share|cite|improve this question

asked Mar 25 at 19:25

SlowLearnerSlowLearner

18513

$endgroup$

share|cite|improve this question

asked Mar 25 at 19:25

SlowLearnerSlowLearner

18513

share|cite|improve this question

asked Mar 25 at 19:25

SlowLearnerSlowLearner

18513

asked Mar 25 at 19:25

SlowLearnerSlowLearner

18513

asked Mar 25 at 19:25

SlowLearnerSlowLearner

18513