.. _sphx_glr_api_gallery_calculus_taylor_series:

Taylor series
-------------

Maclaurin series for standard functions, their convergence, and their
radius of convergence.


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    <div class="sphx-glr-thumbcontainer" tooltip="Taylor&#x27;s theorem expands a function in powers of x - x_0 built from its derivatives at one point, with a Lagrange remainder |R_n(x)| \le M|x-x_0|^{n+1}/(n+1)!. This example draws the Taylor (Maclaurin, x_0 = 0) polynomials of \sin closing in on the function, checks the true error against taylor_remainder_bound, and shows what happens outside a series&#x27; radius of convergence, where adding terms no longer helps.">

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  .. image:: /api/gallery/calculus/taylor_series/images/thumb/sphx_glr_plot_01_maclaurin_series_thumb.png
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  :doc:`/api/gallery/calculus/taylor_series/plot_01_maclaurin_series`

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      <div class="sphx-glr-thumbnail-title">Taylor's theorem: polynomials, remainder bound, and convergence</div>
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.. toctree::
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   /api/gallery/calculus/taylor_series/plot_01_maclaurin_series