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According to a biography of Sri Trailanga Swami, Sri Ramakrishna learned the mantra 'Jiva-seva' from Sri Trailanga Swami, which He later gave to Swami Vivekananda.

Is there any reference of this in the authentic biographies of Sri Ramakrishna?

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As far as biographies of Sri Ramakrishna are concerned, there is no such mention of him receiving any mantra from Trailanga Swami.

In Gospel, it is mentioned that he met Trailanga Swami once and when he did, Trailanga Swami was already following a vow of silence.

A noted monk of Benares whom the Master once met. The Swami observed a vow of silence.

The details of their meeting though is mentioned by Swami Sardananda.

Sri Ramakrishna: The Great Master / As The Spiritual Aspirant / The Master's Pilgrimage And The Story Of Hridayram:

Besides visiting the temples, the Master went to see the eminent holy men of Kasi. Hriday always accompanied him The Master paid a few visits to the famous Trailanga Swami, one of the foremost of Paramahamsas. The Swami was then observing the vow of silence and was staying at the Manikarnika Ghat. On their first meeting, the Swami placed his snuffbox before the Master for his use by way of giving a cordial reception to him. Examining his sense organs and the limbs of his body the Master told Hriday that he bore the signs of a true Paramahamsa and that he was, so to say, the living image of Siva. The Swami had resolved to have a Ghat built near Manikarnika. At the request of the Master, Hriday helped that work forward by placing a few spadefuls of earth there. One day, afterwards, the Master invited the Swami and brought him to Mathur’s residence and treated him to rice-porridge with his own hand.

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