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The number of terms in the sum may be very large or infinite, in which case a partial sum is evaluated directly and the remainder is approximated using integration." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/integrate/_tanhsinh.py", "item_line": 962, "item_type": "function", "aliases": [ "scipy.integrate.nsum" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compute the infinite sum of the reciprocals of squared integers.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy.integrate import nsum\nres = nsum(lambda k: 1/k**2, 1, np.inf)\nref = np.pi**2/6 # true value\nres.error # estimated error\n(res.sum - ref)/ref # true error\nres.nfev # number of points at which callable was evaluated\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nCompute the infinite sums of the reciprocals of integers raised to powers ``p``,\nwhere ``p`` is an array.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy import special\np = np.arange(3, 10)\nres = nsum(lambda k, p: 1/k**p, 1, np.inf, maxterms=1e3, args=(p,))\nref = special.zeta(p, 1)\nnp.allclose(res.sum, ref)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nEvaluate the alternating harmonic series.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res = nsum(lambda x: 1/x - 1/(x+1), 1, np.inf, step=2)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res.sum, res.sum - np.log(2) # result, difference vs analytical sum\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "mpmath.nsum", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "mpmath.nsum" }, "kind": "module" }, "descriptions": [], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "f", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "a", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "b", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "step", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "1" }, { "__type": "SigParam", "__tag": 4030, "name": "args", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "()" }, { "__type": "SigParam", "__tag": 4030, "name": "log", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "maxterms", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "1048576" }, { "__type": "SigParam", "__tag": 4030, "name": "tolerances", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "nsum" }, "references": [ ".. 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