# The function sequences and Taylor series with a fuzzy argument

## Abstract

The main consideration subject is functional sequences *f _{n}(A)* with fuzzy number A for an argument. It is supposed that lim

_{n→∞}

*f*and lim

_{n}(x)=f(x)_{n→∞}

*f*, and these convergences are uniform on each interval within supp

_{n}’(x)=f’(x)*A*. It is also supposed that the equation

*f(x)=y*with respect to

*x*has finite number of solutions for each

*y*on each interval within supp

*A*. The paper proposes sufficient conditions for

*f*to converge in the sense that the sequence of membership functions

_{n}(A)*μ*: converges point-wise. It is proved that lim

_{fn(A)}(y)_{n→∞}

*μ*for all

_{fn(A)}(y)= μ_{f(A)}(y)*y ϵ P*, except such

*y=f(x)*, that

*x*is a discontinuity point of

*μ*, or

_{A}(x)*f‘(x)=0*. As a particular case of sequence

*f*, the generalization of Taylor series

_{n}(A)*f(x)=∑*is considered for real analytical function

_{i=0}^{∞}(f^{(i)}(x_{0})(x-x_{0})^{i}/(i!))*f(x)*for the case of fuzzy argument

*x=A*. Convergence of the series is considered in the sense of point-wise convergence of the partial sum

*μ*, where

_{Sn(A)}(y)*S*.

_{n}(x)=∑_{i=0}^{n}(f^{(i)}(x_{0})(x-x_{0})^{i}/(i!))## References

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